For you what are the best kind of ships? Missile, Quantic? Protonic or ionic???

Well, while playing the campaign, I just kept researching and improving the proton weapons, as the campaign has already given you a decent head start in that direction. I also added some missile research here and there to supplement the proton weapons for some added kick against larger ships and bases. Starting to research in another direction such as Ion or Quantic seems too counter-productive, and will lead to having a few weaker weapons, rather than one (Proton) really kick ass weapon tree (supplemented with some Missile tech). Will be starting to play some skirmish games soon with the Solon Heritage game, and in that, I plan to play around with the other weapon research trees for some different fun :D

You've got enough research points to go down one or two weapon trees fully. ANY weapon tree. The in-game help notes this.

Not that Proton is a bad choice, as such. But you really do have the option.

Based on what criteria? It seems to have some advantages and disadvantages and is pretty well balanced. I've used a single missile battleship against a single ion battleship, and it generally comes down to which ship is higher level.

IMHO the proton is for the "humans". But one of my gameplay I use Quantom from the start and it is also a viable choice.
But. You need the Missile to, if you want to conquer planets fast, and for orbital bombardment the Missle (B) ships is the best.

So, you need to research first the ship tech, and equipment, after it the choosen weapon, and only after it the missile tech.

Ok , you can go oalone to Missile, but it is a really slow gameplay...

All weapons are viable on thier own with the possible exception of missiles.
You have enough research to fully unlock 2 weapon braches on your own.
If you use espionage (for that sweet 90% discount), you can unlock almost everything if you give some thought to it.

Proton
Orange balls.
Part of the lead-in campaign has a directive to research these, so you effectively have a headstart on them. Basiaclly the default Human/Earth weapon.
Decent damage per shot and high fire rate makes them good all-round weapons.
They tend to have less than optimal accuracy though, and will miss targets frequently. The rate of fire makes this less of an issue.
Can eventually be upgraded to pierce shields and bombard more effectively.

Ion
Blue beams.
The default Kariak (and Mars) weapon.
The weakest per shot hit of any weapon, but very accurate. (Usually only ever misses shots when switching between targets.)
Generally better against smaller ships than larger ones, and not great at bombardment.
Would be the worst weapon if not for the special effect they can be upgraded to have: They can DISABLE shields.
It's usually worth having some Ion units around just to proc the special effect and make other weapons more effective.

Quantum
Purple haze.
The default Darzok weapon.
Very high per shot damage, but with mediocre accuracy and fire rate. It does have an area of effect though.
Has to batter down shields the 'hard' way as they have no special shield effects. Better against groups of targets rather than individuals.
Can be upgraded to have an even harder hit and wider area of effect.
Really good when combined with Ion and it's special effect.

Missiles
The Love/Hate weapon.
All three sides have access to missiles, and they all behave the same. Up to a point.
Missiles have the hardest hit of any weapon type. But they are vulnerable to ECM, and travel so slowly that vitually any taget blessed with the gift of movement will be able to avoid them. (Especially if micromanaged.)
They can track targets a little, but not nearly enough to compensate. Better against larger (slower) targets than smaller ones.
Missiles though, are the best bombardment weapon available, and are worth using for this alone.
There's nothing stopping you using missiles alone, it's certainly possible to do. It's more an excercise of frustration rather than difficult though. Low shot speed, lots of misses, everything takes forever etc.

Where missles get interesting is when they get upgrade to the 'advanced' level.
Each race takes them in a different direction, and has different special effects.
Human tech makes missiles move faster, track better and become more resistant to ECM.
Kariak Tech makes missiles penetrate and disable shields.
Darzok tech makes missiles hit even harder, and eventually turns the (B) versions into massive dumbfire rockets intended smash planets.
And yes, with espionage you can combine ALL these effects.

The takeaway from this should be:
Choose whatever you like.
But always be abusing those spies so you can experiment at minimal cost.