I've noticed, especially when cold, I put the car into gear from a dead stop and there's the normal bump of it going into gear and then a second bump a second or so later. With my foot off the brake, it starts to roll when the first bump hits but rolls faster with the second. Do these things sometimes go into second gear on initial engagement to soften things up and then down to first or am I looking at tranny problems already?

 

It's normal. There are a few threads on this if you do a search. The only problem with the tranny is it's lack to cool itself properly so it heats up in stop and go traffic. Then the car becomes very sluggish and won't shift into first.