I have played this game earlier. But I only came (In main quest) where you should "save" a Argonian in Vivec city and then get to the sewers and talk to someone there.

I don't know why I stopped playing really, but I just couldn't continue.

But I think I shall try again. But to you who maby can this game inside-out. How shall you start in this game?

Grind (don't like grinding) Alchemy, sell potions and train your melee/bow skills from a trainer? Or what "shall" you start doing!?

There's nothing you really should do. The game grants you liberties to do many things, and it even offers you ways to solve many things using not only 1-2 skills, but rather 3-4, or more, with some experience and imagination that come in time. Just take your time. Don't stick to only one questline, look around, get to know things, the game. You will find out there's no "right" or "wrong" way to build a character.

The only thing you could need on many quests at the beginning of the game is an ability to persuade NPCs. So either Speechcraft or Charm spells (under the Illusion school) could come in handy. Then again, there are scrolls, and for example a drink called Telvanni Bug Musk that can make you very persuasive even without either of those skills.

Don't grind! One, you don't like it yourself, you said so. Two, one of Morrowind's bigger problems is that you are bound to get "too good" anyway, and grinding gets you to that kind of position even quicker. If you like challenge and want to feel scaled with the game world, do not grind. Rather, make a bad build - if that is even possible! Again, Morrowind offers you many ways around things it throws at you, and it's ultimately up to you to decide what you want to do, in which areas your character rules and in which (s)he svcks. You can get forward in the game (if that's even what you wish) in many ways. No skill is absolutely necessary, or complete waste of resources either.

A few hints.

1) Some skills are very hard to level up. Sneak, Marksman, Hand-to-hand, and all the magic schools come to mind. You need to be good enough in those to make it work. Sneaking even as major skill and in its starting 40s won't likely be good enough against NPCs, so you must pay trainers to train you or practice it on critters like Scribs. Marksman... well, you can hit nothing if you don't take choose it as one of your major skills. Same goes with Hand-to-Hand. Any magicka you want to cast successfully should be chosen as at least minor skill in the beginning, and even then you might have troubles with failed spells and too small a mana pool.

2) Fatigue! Whatever you do, your Fatigue bar must be full, or close to it! Otherwise you will pretty much fail. To hit with any weapon, to cast any spell.

3) Choose at least one melee weapon skill, one armor (or unarmored) skill, and one ranged attack (e.g. Destruction or Marksman) for effective battling.

Clearly the game's variety in different skills confuses you, and you don't know what you should do. Well, anything. You can't do wrong by following those few guidelines, i.e., the hints 2) and 3) in particular.