DT, a couple compliments are probably fine, just don't overdo it as it will sound fake.
Make sure at the end of it, if you want a second date, you should suggest it. Something like, "Let's do this again. Can I take you out to <insert place here>?" That keeps the momentum going, and doesn't leave you wondering when to ask her out again.
I also think it's nice to thank her afterwards (maybe an email when you get home that night or the next day). When my husband and I had our first date, I had to bail a little bit early due to the fact that I was starting a new job the next day and wanted to prepare. On the train ride home, I texted him and told him what a wonderful time I had...mostly to make sure he didn't think I was trying to run the other way.
Most of all have fun!
“Inside every cynical person, there is a disappointed idealist”--George Carlin