In terms of most pleasant to shoot/control.

.380
9mm
.45
5.56
7.62X39
.308

I like 7.62x39.
Fairly light recoil, plenty accurate for average plinking, (out to maybe 200m or so), and not too large (protability wise).

Yeah, I was thinking AK47.

That isn't as much a caliber question, as it is a weapons question.  Obviously, the lower power, the more pleasant, but differing designs will change felt recoil and muzzle jump.  Some early recoil operated Swiss designs featured a 'differential recoil' system.  These guns were recoil operated meaning that the barrel moves as part of the operating system.  In FA mode, the next round was fired while the barrel was still going forward.  Thus the recoil impluse first had to defeat the inertia of the assembly.  Apparently, this massively cut the recoil of these 7.5*55 chambered guns, making them almost subgun like.  

Anyway, I suspect fast aimed semi auto fire (or burst) is more effective in a real combat for accuracy and ammo conservation reasons.  Don't be fooled.  FA is not accurate out of shoulder fired arms, although adequate at CQB ranges.  However, it does serve as a fire suppressio technique.  You'd use all your 7.62*51 really quick if using FA.

Driftpunch is on the right track.

Obviously, in general the smaller the caliber, the less recoil/the more pleasant to shoot and easier to control.  Of course, the weapon design has a lot to do with it too as Driftpunch mentioned.  And you have to decide what role this full auto is needed to fill.  My American 180 (.22LR) is very easy to control and fun to shoot, but a 1919 on a pedestal is also very easy to control and shoot too.  As far as I'm concerned, a tripod or pedestal mounted beltgun is far easier to shoot/control than a shoulder fired weapon; just dial it in with the T&E and blaze away.

22lr.

If your going full auto for fun.  Well, 22lr, 500 rounds for $8.

As opposed to squeeze the trigger, $20.

But full auto 9mm is funn as hell.

AK47 is the worst shooting piece of shit there is. 7/62X39 has hard felt recoil, the gun twists rotationally every shot fired.

Lots of 308 rifle shoot nicer than an AK AK

Well, It also depends alot on the shooter.
I don't think a small-sized girl would go for a 7.62x39, nor would a large framed guy go for a measly .25ACP.

For me, 7.62x39 quite a light recoil, and its plenty effective and accurate for most combat situations. Damn reliable, too.

Pistol calibers are worthless for anything other than CQB and last resort (pistol).

5.56

there's a reason our military is using it

5.45x39 with the AK-74 type brake.

I had a lot more control with the M-16 than I had with the Uzi or the Krinkov, but I think the biggest thing is cost, so I would still take the 9mm.