I can buy a new (to me) car now, because I just got a pretty good new job. I will be spending about 13k on it. Insurance for the A4 is between 450-550/6 for me, so I'm pretty open to an A4.

My question is basically, is the B5 or B6 more reliable? I will be traveling quite a bit.

 

Depends on who you talk to and more importantly how well the previous owner of the car took care of it.
My B5 is just under 200K miles and has never had a major breakdown. Audis are not as reliable as Japanese cars and lots of small things can go wrong.

Most important thing is to have a mechanic who specializes in VW/Audi check the car over before you buy it. Plan on spending 1000 dollars in the first few months swapping out all the fluids and taking care of things the previous owner neglected, and keep 1000 in the bank for emergency breakdowns.