One of the best points about Halo's combat system is that you can only carry two weapons at once. This adds an interesting and enjoyable tactical aspect, forcing you to think about what combo to use for a given situation. A few combos are worth a special mention I think, as follows. In each case, the first weapon listed is the one I'd normally be carrying as primary.
Pistol and plasma pistol
You can't go far wrong with this pairing. The pistol is superb on its own for taking down enemies from distance, keeping you fairly safe. It's also a great Hunter killer of course, and is the best thing for bringing down Banshees. The plasma pistol is handy for taking down Jackal and Elite shields, potentially opening the way for a pistol headshot to finish off; that's where the weapons really work as a team. It's also great for melee attacking and close-up work in general.
Plasma pistol and plasma rifle
A power combo, excellent for relatively short-range combat. A plasma ball is good for stunning Elites or taking down shields, and you can switch to the rifle to finish them off. Rapid fire from the plasma pistol is also devastating in itself; more so than the plasma rifle, though the shot speed is lower. If one weapon is starting to overheat, just switch to the other to keep the plasma flowing!
Plasma pistol and needler
This is my favourite combo for sheer vindictive fun, especially in clearing areas of Grunts and Jackals. Pile in aggressive streams of needles to take out enemies explosively. When you come up against a Jackal, take down his shield first with a plasma ball, leaving the way clear to pile a stream of needles into his gnarly hide. Pure bliss.
Pistol and sniper rifle
The ultimate precision combo. When it comes to safety, distance is your friend; and this combo will let you pick off enemies from as far away as possible. In regard to vehicles, the pistol is great against Banshees, and the sniper rifle can pick off Ghost pilots from miles away.
Pistol and shotgun
A great anti-Flood combo. Most of the time you can pick them off at distance with the pistol. Switch to the shotgun when things get more up close and personal; a nice contrast.