Right up my alley. I redpointed 5.13a outside the other weekend. Any particular questions?
Boots actually aren't a horrible idea because they typically have stiff soles. Their weight and general bulkiness would be quite detrimental though. I can actually climb pretty decently in a pair of Chacos (sandals) because they are relatively stiff. At the same time, my climbing shoes are quite soft and sensitive, but with very sticky rubber.
Otherwise, I definitely recommend renting a pair of shoes from the gym. They will absolutely help out because of sticky rubber and a generally smaller profile than a regular shoe, allowing you to better use the holds on the wall.
Keep your arms as straight as possible for as long as possible. This serves two purposes:
1. Your weight will be supported mostly by your skeleton and it takes much less energy to hang on with straight arms than with bent arms.
2. it puts as much weight as you can onto your feet. You should never feel stretched out in that you are on your tiptoes reaching for the next hold - if you find yourself in that position, move one or both of your feet up to a higher foothold, and then move your hands up. In the ideal situation, this is the only time where your arms will be bent.
Try not to get caught up and hang onto the holds for dear life. If you get stuck, don't be afraid to just hang there - otherwise you will get burnt out extremely quickly and you will have a very short session. If you do find that you don't want to hang on the rope for whatever reason, remember to keep your arms straight.
This might be difficult at first, but try and determine a sequence for the particular route or problem that you will be on. Typically for beginner climbs, there will be a rather obvious right/left/right/left sequence to the holds.
Don't forget your feet. Good footwork and technique are what separates good climbers from the "bad."
One more thing, the gym you go to will probably have their climbs rated on the
YDS system. Anything below 5.7 will be a good initial climb to get on, and you can progress from there depending on how you feel. If you are bouldering, (scroll down on that page) you will want to start on V0 or VB if they have it. You will find that each individual grade on the YDS scale is much easier to jump than a grade on the bouldering V-scale.
Just as a FYI, any particular gym in this country (outside of Boulder, CO) will probably have grades between 5.6/7 to 5.12d or so, and V0 to V8-10.