Depends on your dose. I did a week on 25mg, then moved up to 50mg where I have remained. I experienced no side effects except I recommend avoiding alcohol (or drinking only in moderation). The hangovers are amplified x 100 and I think it is dangerous for your liver. I also find if I overdo things (such as nights out with friends, even if I am not drinking) I seem to come down with illness very easily and a cold that in the past may have lasted a day, would take a week to clear.
Most people experience nausea when they take azathioprine but this usually passes and I was fortunate enough not to experience it at all (probably due to my low dose).
Symptoms to watch out for are extreme pain in the centre of your abdomen - a rare side effect called pancreatitis, which requires immediate hospital attention.
Make sure you get your required blood tests (usually weekly for a while, then every two weeks, then every month, then every 3 months etc). This is to check the medication isn't making you unwell.