Ok, I am absolutely not a transmission expert, but there is a lot of tranny/ls1 info on LS1 tech, including a pretty informative discussion on your T5: http://www.ls1tech.com/forums/manual-transmission/137289-ls1-t5.html
Second, there are a number of transmissions that seem to be "the" option, depending on your desired uses. If you want to stay automatic, and use it for general purpose and highway driving, the 4L60E is popular. Straight line track stuff people seem to stick to the TH350. Standard choice is generally the T56 6 speed.
Personally, I am in the process of shopping around for a transmission rebuild place that wants to sell me a rebuilt T56 and give me some swap credit for the TH350 that I have. My general tactic has been to call the shops, tell them what I have, and what I am looking for, and ask what their price would be. Some places will work with you, some won't, but I stick to specialty shops because they generally know what they are doing. I'll pick a steering arm or a door out of a junkyard, but a tranny is a different beast for my level of experience.