You almost certainly do not want a "mountain bike".
A "mountain bike", for me, has:
- quite thick tyres
- lots of gears
- potentially a thick/heavy frame
If you're riding round town, those thick tyres will just add friction to your ride. They made it harder to pedal, they make you slower, and you don't need them.
Unless it's very hilly where you live, you don't need 21 gears. You might need 7, tops. (I live in Melbourne, where it's very flat. I have 1 gear.)
You almost certainly don't want or need a thick, heavy frame. Same reason you don't want thick heavy tyres.
I'd look at some sort of hybrid: not super-skinny tyres like the racers use, but nothing with "knobbles". You only need knobbles if you're riding through mud. You won't be. A hybrid is more likely to be built with a basket/panniers in mind; they're designed for commuting.
And, don't worry too much about "rough road". Unless it's really rough, any decent modern frame will handle it. My single-speed with fairly skinny tyres handles anything I choose to ride over.
I see so many people with mountain bikes in the city and none of them look like they're having a good time! Google "hybrid bike", you'll get results for your city. Go and have a look around some shops, talk to the staff, take a few bikes out for a ride.
Good luck! And have fun. :-)