A front suspension MTB can be a reasonably good all-round bike that is good on trails.
To enhance its road-going performance you may want to fit less knobbly tyres, the slicker the faster.
Ensure that the frame is the correct size and comes with threaded eyelets for a luggage rack and fender.
Discount no-name bikes are generally useless. The entry-level bikes from respected brands are quite good.
The first step is to pick a good local bike shop, one that you trust and where they have good advice about size and accessories.
Budget for the bike + helmet, simple repair kit.
For utility riding you may want to fit a luggage rack and fenders.
A road bike is much faster on the road but is hard to ride on trails. Race-oriented road bikes are usually unsuitable for utility riding, they are geared too high, have little tyre clearance and cannot take luggage. Tour-oriented road bikes are much more useful and can even take some trail riding.
Using the bike for commuting and shopping is an easy way to fit excercise into your everyday routine.