You shouldn't "exercise" immediately after surgery. While you're healing you want your stitches to stay and you do need that energy to rebuild. But not doing high intensity stuff doesn't mean stay sitting down. Moving and light stretching is important. Definitely walk around.
I'm told the amount of exercise you do through chemo depends on how your body handles the treatment. If you have no energy from doing nothing your body needs to be resting, so don't push it but keep moving to stave off stiffness. I can't tell you more than that I'm afraid. I'm sure someone who's been through will chime in.
A personal perspective -
This will sound a little defeatist, but for someone who hasn't started going to the gym and isn't set into a routine already, I'd recommend taking the $400-ish you'd pay for the yearly gym membership and investing it in a treadmill/cross-trainer/exercise bike/your preference of light exercise capable equipment.
Turning going to the gym into a habit is difficult enough without running into reasons not to. It happened to me in my first year of uni, I signed up and started going regularly when "I have to hit this paper/study for this test/exam/do this research etc." started and suddenly it was three months later and I'd only been twice. I imagine for you it would be "I'm too nauseated/exhausted/in too much pain" and suddenly it's November and you wish you'd just got a treadmill. Haha!
I only had a small operation, but almost a month later there is still pain enough for running to be out of the question for me. I know chemotherapy is going to hit me like a tonne of bricks, I already have unexplained pain in my everyday and that will only get worse. So I turn on my machine every day and walk whenever I have a spare moment to make a habit out of moving. I'm writing from it right now - wearing my pajamas and in a pair of socks plodding along. It's just here in my house, available in a moment with no going outside and facing the world preparation needed.
Swimming is the best!! Now I wish I had a pool in my house too - I don't even have a bath as a consolation prize. hahaha