Unless you had other uses for the air compressor (you will, once you get one), I would forgo that purchase right now.
I have 3 grease guns: a small (uses small cartridges) single-hand pump, a large (uses full size 14? oz. cartridges) and an air operated one.
If I'm doing a routine maintenance greasing like all the fittings on the loader, backhoe and tractor, then I'll usually use the air power grease gun. I like this one 'cause I can hold and operate the gun with one hand and keep the flexible hose firmly attached to the zerk fitting with the other.
If I just want to give a shot of grease to a single fitting or two, I'll usually use the small gun. This has a rigid pipe, but the gun is small enough to get into most spaces. However, the small cartridges are more expensive, so this isn't as cost effective IF you are going to do a lot of greasing.
The problem I have with the larger hand-held gun is that it takes 2 hands to hold and pump, and with a flexible tip, the end sometimes slips off the zerk and makes a bit of a mess. If I put the rigid pipe tip on it, it's not so bad, but there are sometimes clearance issues with some fittings and I can't get the gun in a good position to seat the end over the fitting.
Oh, and I know I could change tips on the guns, but it's easier to just grab the gun with the right tip already on it.
If I had just one gun, I don't think I'd want it to be air operated, 'cause then you're dependent upon the air supply everytime you grease. For me, it's not always as convenient.
The battery operated greasers look real nice, but I have to agree with Bird, they seem to be over-priced. If they were $40 - $50, it might be worth it, but I think they're in the $160-$200 price range. And again, you're dependent upon an additional power source (a battery) to do your greasing. I'd still want a manual one for back-up.
Oh, and no mattter what type of gun you get, I recommend that you wipe the zerk fittings clean BEFORE greasing and AFTER greasing.
If you decide to get an air compressor, get one that can operate real air tools, such as an impact wrench, sand blaster, etc. It probably won't cost that much more and you'll then you'll find lots of uses for it.
Good Luck,
~Rick