halpx

So I decided I need one, for the three primary purposes:
* Getting to my university. It's five kilometres, mostly concrete, although there's a short foresty patch on the way if I don't want to go around.
* Having fun around here. I live in the outskirts of a big city, but here there's just one concrete road, and grass/fields/forests all around. I'm planning long, several-hour trips.
* Getting to the shop. A regular, solid road.

I was in a bike shop and told the guy there about those, and he showed me a bunch of bikes, all the same to my inexperienced eye. So I ask you for any tips, what to look for given the above conditions, etc.

Bikewer

The diversity of the modern bike market can be daunting to a new customer.

Think of three basic types.... Road bikes, with the skinny tires and "drop" handlebars. Meant for riding fast and far on pavement. Mountain bikes, with (usually) gnarly knobby tires, suspension, and lots of gears...Meant for off-road use but fine for all sorts of cycling jobs.
And finally the hybrids, which often have "mountain" components but "road" frames and intermediate tires. These are happiest on pavement but can be ridden off-road on paths and improved trails.
There are subdivisions in all of these things....

If I were you, I'd be looking for what the companies are selling as a sort of "urban" mountain bike.
A pretty straightforward mountain-type frame set up with street tires. These bikes are very practical, ride decently on pavement, and yet can handle bike paths and light trails pretty well.

stapfam

Bikewer has it right. Various bikes for various jobs. If you want offroad then go for the Mountain bike. Just unmade trails and road then the hybrid and it doesn't sound as though your area is right for a road bike. But a bit of shopping- try to get one that will take a pannier rack- not all bikes will- and try to decide what type of riding you will mainly be doing.

And get one that is the right size and not just one the shop want to offload.