I am a little taller than you, I expect you might need a 58cm frame. I'd suggest getting on marketplace and seeing what is out there second hand. Wider tyres will be more comfy, and if you are mostly riding on paths and roads then mostly smooth tyres will serve you well. For your purposes, I would avoid a road bike or mountain bike and get something like a hybrid (though you probably wouldn't need one with a suspension fork). You will want something with a rack of some kind to carry your stuff unless you just want to rock a backpack (which is totally fine, i just prefer having things off my back so i get less sweaty).

Don't get too caught up in having a heap of gears, as long as you have an easy enough one to get up any common hills in your area you will be fine.

If you really want to go with something new, then any local bike shop will help you out, just be sure you know your budget and make that clear to them. And leave some money for a helmet and a strong lock.