Theophilus,
Regarding sharing successful strategies, I guess I have noticed a couple of patterns that seem to work okay, but usually my objective is to just plant the seed. I feel like there are probably some questions that people have always had that are unresolved. I generally try to focus on those. Here is probably way too long of an answer to your request to share strategies.
Background
For die hard Catholics, all I can say is be careful where you step. I know this because I was one. It is a bit of a force field and sometimes information that comes up. Most of it stems from one key word - authority. This comes from the myriad of protestant offspring and this person will likely have concluded that the RCC has a monopoly on our Father because of the indoctrination, years of tradition, and misinformation. To better understand this, it may be helpful to understand where I came from. I was in RCC school for 13 years of my life. I never bothered to pick up the Word and read it except for Revelation and a few bible studies here and there with a deacon that I was fond of. Anyway, as I started a family I became concerned from my kiddos upbringing, etc. so we started going back to church. There was deacon that I had always been a big fan of, who held bible studies, etc and would often ruffle feathers in the local diocese. He would do his homily and say that all the traditions in the RCC meant nothing if you didn't have a relationship with our Creator. This was the one point he kept trying to get across. Well one day he offered to do a study of the Prophets. He basically walked us through some of the anger spells in the Old Covenant - I was thinking man, this God is one angry dude, but then.... we got to Jeremiah 31:31 and all of a sudden a seed was planted. I will no longer remember their sin. It was the most poetic ending to a story I ever heard.
So I decided I needed to do my own Bible study. I started from the beginning, by the time I got to Yah'shua's assessment of the Pharisees I was in tears for days. Then I started to see some of the patterns, but I still had that nagging question.... what is the "one true church" and it seemed like the RCC was the only likely candidate given that there were so many differing theologies out there, and it seemed to have the longest history and in my mind at the time, and probably had much more time to study and understand everything than some of the newer churches.
After that, I started trying to "prove" to myself that the RCC had authority and then I wound up proving the opposite. Then sometime around last October I started a search for the "One True God" and the "true meaning of life", stuff like that. I realized the questions I had been asking all my life were the wrong questions. It wasn't what was the one true church, but rather Who is God?. That there was a creator, etc. I never really had a problem with, and I don't think you'll have this struggle with most RCC, but what you will have is the taking of selected verses, but never really getting into the subtle details.
Then I found myself in a Bible church as my marriage was struggling. They went over Song of Solomon in excruciating detail and I loved every minute of it. There were things in there that I never knew. Through my own efforts many parts seemed quite boring. But once that dissection of the words occurs, it just felt like a an irresistible draw.
Paint the Word in a New Light
So the question becomes, what was it that got to me? Partially our Father by allowing my marriage to hit some snags perhaps, but I also think the uncovering of Wisdom that I had never known was in the Word revealed through the analysis and uncovering of details. The key word being never known. You can read it all day long I think and it may just sound like a story, but when subtle details are presented, that make sense, that add up, well....
So how do you apply this to other RCC? Well, some of them get it I think. Some of them even call him Yahweh believe it or not, but if you can show them things that they never knew were in there.... well, that's how you breech the shield.
So how do you break the ice? Well find an interest point in the conversation... the Iraq war, History, the differences between various languages... then slowly transition into the differences in the languages behind the scriptures, and talk about some of the things you've found almost matter of factly, but not shoving it down their throat, if you do that, congratulations, you may have just helped to destroyed a soul, perhaps not culpable for it, but none the less, a scary position to be in.
Prayer / Teeny Tiny Miracles
Now there is another tactic involving just basic prayer, but I have to admit I may be getting prayers answered because of my new found faith or something. I've had what I consider many minor miracles, no parting the waters mind you, but things that just seem too "out of the blue" to be coincidence. The challenge in this area is distinguishing between coincidences and what seems to be answers to prayer. Personally, I've found that the answers to my prayers tend to line up in either a series of weird coincidences, or the timing is just impeccable. So what I do is tell the person that I'm going to pray for something, but I'm not sure if our Father will answer or not, but we'll see and you can make your own decision if you think I'm off the deep end here. For example, my brother and his wife had been wanting to have a kid for about two years, but were not having any luck. They were planning on going to fertility doctor, but hadn't quite gotten there.
I told him I'm still learning, but we'll see if our Father will place some favor here. So I said let's try to do what they did a long time ago in the scriptures. I laid my hands on him, kinda the way they write about it at least in English translations, and said a silent prayer from deep down inside somewhere. Lo and behold, he calls me up one month later (the time required for her cycle to have passed). They were with child. Now when I prayed, I basically said, I'm not trying to test you or do this for me, but if it will help spread knowledge of you, can you grant a favor with my brother here. Then when it happened, my brother started to believe. So... that is one way I guess.
Planting Seeds
Most folks I run into, believe in a God of some sort, it just seems to be common sense, but with atheists and the like, that is a whole different ball game. What I've come to realize is that all of us have our own little decision trees and the the quality of the information in that tree will be biased in a certain way depending upon a person's course through life. What I try to think of it is, what seed can I plant today, that will get watered down the road? Usually language is my angle of attack so far. So they'll make an argument is based on English translation. I'll say, umm, ohh.. you're using an English version, well no wonder... and just move on. Plant the seed. As long as I can plant the seed, I figure mission is accomplished, because otherwise you'll sit there debating all day and potentially drive them further away, as I had to learn the hard way.
Now where am I struggling, well with "Christians" that seem to understand, but are reluctant to let go of Christmas, etc. These are my family members. What I've gotten so far, is well, "don't you think that if Jesus thought that was important he would have straightened them out back then" and a serious of erroneous statements of fact based on what they were taught. This is perhaps the most frustrating.