There are various resources out there that explain why the Church believes so strongly in traditional marriage and how the Church loves every person out there. You are not alone in your acceptance of Church teaching, but to evangelize the culture you have to understand your faith. These are some resources that I know of; I recommend you take some time and look them up.
First of all, the Catechism teaches in paragraphs 2357-2359 on homosexuality. You'll find that the Catholic Church hates no one and teaches hate towards no one. The Catechism teaches about acting on same-sex inclinations, that "homosexual acts are intrinsically disordered." But it also teaches about the persons who experience same-sex attraction, that "Every sign of unjust discrimination in their regard should be avoided." Notice: Church teaching distinguishes between the act and the person. Acts are sinful or virtuous. Persons are beloved children of God. The inclination that a person feels but didn't ask for is not a sin, only actions are.
I've linked to it before, but I want to point you again to the homily that my friend Deacon Andrew Kinstetter preached the weekend after the Supreme Court decision. I think it's the best homily he's preached in his short time as an ordained minister. He showed the beauty of marriage, and that it is beautiful because it gives life. Marriage unites and marriage creates life. Marriage is a gift from God to help us reach eternal life. By showing the beauty of marriage, he showed quite simply how homosexual "marriage" cannot participate in the life-giving power and cannot image the relationship between God and his Church. Do read his homily.The Third Way
This is a 38 minute video that features Catholics who experience same sex attraction and have chosen to live chastely and experience the fullness of life. They boldly tell their stories about how they tried the world's version of happiness and never felt satisfied, and how they got out of that culture and found their way (for some of them, found their way back to) the Catholic Church. It also features some of the most popular Catholic speakers and evangelists today: Sr. Helena Burns, Chris Stefanick, Jason Evert, Christopher West, Fr. Michael Schmidt, and others.
Desire of the Everlasting Hills
This is another video, about an hour long, that has three people telling their stories of trying to be happy living how the culture told them to live. One of them was even in a same-sex relationship for 25 years. But like the other video, the culture's promise of happiness never played out and they found their way to the Catholic Church.
Catholic Stuff You Should Know Part I and Part II
This is a two podcasts from former classmates and seminary friends of mine. They excellently and tactfully explain the topic, and they took the time to address it in two parts; both are about 25 minutes each. One of the most excellent points they make is that we (we the Church, we the proponents of traditional marriage) cannot advance a logical argument for traditional marriage as long as we continue to utilize contraception or as long as we live together before marriage. If heterosexual unions exclude the procreative or unitive ends for which marriage is made, then you can't advance a logical argument against homosexual "marriage." The second part deals largely with pastoral care for those who experience same-sex attraction and calling all people, no matter their sexual orientation, to holiness.
This particular part of Jason and Crystalina Evert's overall Chastity Project deals with homosexuality. A good collection of short videos from knowledgeable Catholic speakers.
This is a solid Catholic organization that seeks to support those who experience same-sex attraction, their families, and their ministers. Their goals (from their website) are chastity, prayer and dedication, fellowship, support, and to be a good example. Not all "catholic" same-sex attraction support groups are good. Beware of the group called Dignity. They misinterpret the Bible and the teachings of Jesus to defend the LGBTQ movement.
Gay Marriage: Brought to you by Contraception
This article advances an argument that was referenced in the podcast above. If marriage is in a bad spot today, it started with the widespread acceptance of contraception. It began with the Anglican Church's 1930 Lambeth Conference. Not that marriage was perfect before 1930, but that's the first time you can point to a thing that was detrimental to marriage being called good.
Heather Does Not Have Two Mommies: Talking to Your Kids About Same Sex Attraction
This is an article from a mother of four who has decided she needs to start talking to her young kids about SSA in the culture. She realizes she can't shield her kids from hearing about it in the culture, so she wants them to hear the truth from her before they hear the lies from the culture, and she offers tips on how to talk to kids about this tricky topic by making sure they understand solidly what marriage is for.
False Enlightenment at the Court
Here is a more intellectual argument (still very accessible) about the hubris present in the Supreme Court's decision. The Court has misunderstood essential and accidental in their redefinition of marriage when they said that procreativity is accidental and not essential to the nature of marriage. Read the article for more.
A small historical analysis of why and how the gay rights movement has won such victories, and which "movement" is destined to win in the long run.
The Church and the New Normal
This article by George Weigel also analyzes how we got to this point and where we go from here.
Here are three books by authors who experience same-sex attraction. I haven't read them, but I trust the brother priest who recommended them. I hope to read them soon.
-Beyond Gay - by David Morrison (from Denver, forward by Archbishop Chaput)
-Sexual Authenticity - by Melinda Selmys
-Washed & Waiting - by Wesley Hill (protestant author)
Those are the highlights of what I know right now. I have some other ideas that I would like to reflect on and post on, so hopefully more at a later date. If you know of other resources that might be useful please let me know. Overall, be loving, be genuinely charitable, but be honest. There is no place for hatred or lies in this discussion with our culture, but then again, there is no place for hatred or lies any place in the Christian life.