To begin with, please read Romans chapter 1. I'll be using the Authorized King James Version throughout. As editions do vary, I'd recommend using the KJV now. Later you can check the newer paraphrases if you'd like.

Our CONTEXT begins at 1:15 and 16:

"So, as much as in me is, I am ready to preach the gospel to you that are at Rome also.
For I am not ashamed of the gospel of Christ: for it is the power of God unto salvation to every one that believeth; to the Jew first, and also to the Greek."

Paul, through inspiration of the Holy Spirit, begins the topic by stating that he is talking about the Gospel of Jesus and how it must be preached and practiced, and how those who love God will accept it joyfully. We note, for the purposes of this study, that lesbian, gay, bisexual and transgendered (LGBT) Christians do accept the Gospel both readily and joyfully. This study exists to demonstrate that neither the Bible in general or Romans 1 in particular prohibits this community from accepting and living the Gospel fully.

Verse 18:

For the wrath of God is revealed from heaven against all ungodliness and unrighteousness of men, who hold the truth in unrighteousness;

The wrath of God is revealed from Heaven onto those who hold this truth in unrighteousness. We are therefore speaking of people who hold God's spiritual Truth and salvation in unrighteousness, i.e.. we are discussing abuses which are done in the name of religious truth. Devout Christians do not knowingly do this regardless of the sexual orientation.

Verse 19:

Because that which may be known of God is manifest in them; for God hath showed it unto them.

"Because that which may be known of God..." Again, religious truth is the topic, not sexual orientation or even sexual morality. 'That which may be known' of God' is that which has been revealed in the Holy Scriptures and by the Holy Spirit. These things attest to God's righteousness and should be honored.

Verse 20:

For the invisible things of him from the creation of the world are clearly seen, being understood by the things that are made, even his eternal power and Godhead; so that they are without excuse:

From what is visible we should know the invisible, i.e. God and God's truths are revealed through our senses. We can behold the handiwork of God all around us if we but look and these things confirm the existence of God, hence we have no excuse to deny God's existence.

Verse 21:

Because that, when they knew God, they glorified him not as God, neither were thankful; but became vain in their imaginations, and their foolish heart was darkened.

These people being discussed knew God but chose to submit themselves to their "vain imagines" rather than to God. In other words, they invented new forms of religion, forms that satisfied their base urges and ungodly desires. We are still discussing religion here not sexuality.

Verses 22, 23:

Professing themselves to be wise, they became fools,
And changed the glory of the uncorruptible God into an image made like to corruptible man, and to birds, and fourfooted beasts, and creeping things.

These people claimed to be wise in their spiritual beliefs, and yet they were foolish, rejecting the true God, they worshiped false idols and "changed the glory of God into an image..." and worshiped that rather than the true God. Still, we're talking about religious practices and abuses, not sexuality. The Roman Empire, which would include Jewish Palestine, was filled with such worship. Pagan temples offered everything desirable and blasphemed the Creator of heaven and earth. Paul here condemns such worship.

Verse 24:

Wherefore God also gave them up to uncleanness through the lusts of their own hearts, to dishonour their own bodies between themselves:

"WHEREFORE..." Wherefore, i.e. because they rejected God and worshiped idols and followed the paths of their vain imaginations rather than worshiping the One True God... FOR THIS REASON God gave these people up to their lusts.

The CONTEXT of these lusts is the unauthorized desire-based religious practices of the day. These have nothing to do with modern day consensual sexual relationships, whether they be heterosexual or homosexual.

As was common throughout the very religious Roman Empire, such idol worship INCLUDED sexual 'sacrifices' and rites wherein the worshiper engaged in 'sex magic' with temple priests and/or priestesses in the guise of spiritual ecstasy -- AS THOUGH the temple prostitute were the Pagan deity incarnate. THUS, the context here has NOTHING to do with contemporary homosexuality among Christians (or otherwise).

Verse 25:

Who changed the truth of God into a lie, and worshipped and served the creature more than the Creator, who is blessed for ever. Amen.

"WHO..." i.e. these people engaging in ritual sex magic and worshiping the Pagan idols, who were "exchanging the truth of God for a lie," and who were committing idolatry through their sex rites and thereby serving the created images and their own carnal desires rather than the true God, who alone is blessed for ever.

Verses 26, 27:

For this cause God gave them up unto vile affections: for even their women did change the natural use into that which is against nature:
And likewise also the men, leaving the natural use of the woman, burned in their lust one toward another; men with men working that which is unseemly, and receiving in themselves that recompense of their error which was meet.

"FOR THIS REASON..." i.e. because these people exchanged the worship of the true God for idols and sex rites, God "gave them up" to their own carnal desires.

The natural use or purpose of sexuality according to the Bible is procreation AND building/sustaining strong marital relationships (comp. I Cor 7:2). These people however were using their sexuality "against nature" by employing it to worship the idols of false gods. Through such sexual activities they sought two basic goals: One was the physical pleasures of sexual congress and the other, perhaps more significant in this context, was the promised spiritual union with their gods and the acquiring of benefits thereof. This is not the goal of sexuality intended by God but as they people willfully rejected God, they were allowed to go their own ways. Again, this was very common at the time throughout the empire.

These worshipers were "working that which is unseemly." The use of the word "working" shows that there was a specific end result in mind, i.e. earning the blessings of the Pagan deities, the normal reason for all idolatry. All they received for their "work" however was venereal diseases "...and receiving in themselves that recompense of their error which was fitting..." The sex cults throughout the empire were hotbeds of sexual diseases and the worshipers often became infected.

THOSE WHO TWIST these scriptures by saying they refer to HIV in our contemporary culture -- and I know some of you are out there! -- need to consider that this verse is the only one in the Bible which can be TWISTED to include lesbians. Lesbians remain among the lowest at risk for HIV and, after years of education, the gay male community is no longer the highest at-risk group. If this dreadful illness were God's wrath against homosexuals neither would be the case, nor would children be so at risk, nor the disabled, nor those in need of blood transfusions etc..

Verse 28:

And even as they did not like to retain God in their knowledge, God gave them over to a reprobate mind, to do those things which are not convenient;

"And even as they did not like to retain God in their knowledge...." The subject is still religious faith; it has not changed. These people did not want to acknowledge the true God's existence. They preferred to worship idols of their own imaginings and trust in dangerous sex rites for their salvation. AGAIN AND STILL the subject matter, the context, is not homosexuality but idolatry coupled with sex rites, heterosexual AND homosexual. FOR THIS REASON, because of their spiritual harlotry, God gave them over to a "reprobate mind" and allowed them to believe that their false worship was approved by God when in reality, they were in direct opposition to God's holy will due to their pseudo-spiritual abuses.

In sharp contrast with this, LGBT Christian ministries are growing like wild fire, individual independent churches, groups like the Alliance of Christian Churches, the Gay Pentecostal Alliance, The Evangelical Network, the Universal Fellowship of Metropolitan Community Churches and so on, are now home to thousands of sincere homosexual followers of Jesus who WANT God in their knowledge, their minds, their hearts and in their lives just as other Christians do. In addition to these ministries, more and more mainline churches and denominations are becoming "reconciling and affirming" of LGBT people. This fact alone demonstrates that these verses are not referring to gay and lesbian Christian people. As with any other community, lesbians and gays vary as to their acceptance and understandings of the Gospel message. A very large percentage of those who are not pro-Jesus have had their hearts hardened by abuses which well meaning Christians have subjected them to. The Church is failing miserably at evangelizing this community and will doubtless be held accountable by the one who said "Whoever comes unto Me I will in no case reject."

Verse 29-31:

Being filled with all unrighteousness, fornication, wickedness, covetousness, maliciousness; full of envy, murder, debate, deceit, malignity; whisperers, backbiters, haters of God, despiteful, proud, boasters, inventors of evil things, disobedient to parents; without understanding, covenantbreakers, without natural affection, implacable, unmerciful:"

What resulted from their being "given over to a reprobate mind?" As a result of their religious rebellion against God, these developed the various moral weaknesses described in these verses. LGBT Christians, as all other Believers, seek to avoid such traits in their lives.

Verse 32:

Who knowing the judgment of God, that they which commit such things are worthy of death, not only do the same, but have pleasure in them that do them.

"Who knowing..." i.e. the people being discussed in the previous verses, these idolatrous worshipers who are filled with all unrighteousness etc., "knowing the judgment of God, that they who commit such things are worthy of death..." That idolatry merited the death penalty for Jews under the Mosaic Law is clear. As I explained elsewhere however, despite the two verses in Leviticus which are sometimes cited, homosexuality is not a capital offense nor even negatively commented upon in the Torah.

CONTEXT IS ESSENTIAL!