A rather simple game engine to look at is Love2D - I know you wanted 3D, but my main experience is over there. Really, the gist of an engine will be about the same either way.

An engine mostly handles a few basic things like device input, managing the screen (with niceties for things like buffers, sprite maps, etc), drawing, and sound.

Slightly more game-oriented are things like managing the game loop, having an update timer, or calculating collision detection.

3D engines quickly help to smooth out the frustration that is working in 3D in general - lighting, cameras, etc.

As someone else said in another reply: if you want to make a game, see if there’s an engine that works for you. If you want to make an engine, or have very specific needs that require the flexibility and optimization of a custom engine, do that.

If you're interested in making and finishing a game, there is no good argument to be made for writing your own engine unless you have some very very specific rendering requirements (e.g. you want your game to be in hyperbolic space or something, or have 3D portals).

In every other case by writing your own engine you'll be wasting your time on a million already solved problems instead of actually, you know, making a game.

If you're interested in game engine development, or graphics programming, and not so much in finished games, then you can write some engine code as an educational exercise, though.