If you don't know where to start, then the game you are proposing is probably too advanced... I am a beginner, too, thought. How many games have you made before this?
A game engine is a set of functions that lets you draw to the screen, and other basic game functions like that. If you write your own engine, you will basically be writing functions that use DirectX or OpenGL and you will be calling those in your game. If you use somebody else's engine, then you will be calling functions that probably do the same thing, but you wont have written them. There is always the chance that the engine you are using will slow down performance of the game though.

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I have done some (basic) programming against DirectX but I have troubles with collision detections and handling walls (in multiple rooms).
It would be great if there was a game-engine that you could use that handles collision detections, maybe gravity, bsp(?), multiplayers plus had an acceptable frame-rate :-)

Doing collision detection is basically the same between 2D/3D games. The only difference in doing it in 3D games is the added demension. In the 2D games, you just have to test squares/rectangles against each other for simple sprite collision detection, and the math for this is easy. The math gets harder in the 3D games because you have to use 3 demensions and then test the polygons against each other which would take a lot more calculations. If anyone needs help trying to figure these out just ask, I've tried the 2D collision detection and it works, but I havn't tried the 3D collision detection.

There are plenty of engines out there that would do what you're looking for - Genesis3D is one that pops to my head. Look in the Links section of this site.

Writing 3D Collision Detection that works is really easy, as they said, its almost the same as 2D Collision Decection, The HARD part is to make your collision detection FAST, and if yo use someone elses engine for that I promise it will not be as fast as if yo wrote it yourself, because yo know what yo want it to do, and what kind of Collision, etc....

If i want to develop the following game, is the best (quickest) way to use a game engine? :
* Network playing (Up to 50 users at the same time)
* Indoor environment
* Somewhat like quake-interface

I have read a bit about DirectX, OpenGL, Genesis3d but Im not sure what to start with..
Anyone got any good advise?

Not to discourage you but if you have difficulty with the aspects of 2d games you shouldn't be proceeding onto 3d quite yet. Don't start off trying to make quake, not even the technological weenies at Id software came out and made a 3d game to start off with. Keep pluging away and when you are ready to proceed onto 3d, look into Genesis or another free game engine, they are actualy non-API dependant, you can use D3d, OGL, or GLIDE just by writing a driver for the engine to use. And genesis also gives you all the tools you need to make a game, model importer/map maker/texture compressor/ etc.

Have Fun.