You are wrong there on what you said about caps and batteries, alaskanzx5. Caps do indeed store power, but the power is depleted within one second. But batteries do not store power, they create it. It has to do with the plates inside the battery that create the power. Caps are mostly only beneficial because people put them next to their amp in the trunk or wherever. That allows it to draw more quickly from the cap than having to travel all the way to the front battery/alt.
And as far as it being about power(watts), it also depends how efficient your amp is. You can run 600 watts off a 4 ohm load and have no problems what-so-ever from your amp. The reason being that your amp is a lot more efficient at that load than at a lower one. If you have 600 watts off a 1 ohm load, your amp is less efficient, therefore drawing more current than before to get the same output of 600 watts. A lot of that power lost transforms to heat. That's why you feel your amp getting warm/hot at lower impedance loads than at a higher one. (I'm saying this as if you had the same power output at each load.)
I would take the same steps everyone else has mentioned already. So start with the "big 3" upgrade. If you still have problems, check your grounds, or add a few extra under the hood. The next step would be to invest in a new starter battery for under the hood. You can check out Optima batteries or anything similar. Those tend to work good, and also have a nice warranty if anything goes wrong! If you don't get an optima, try to shoot for something with a high AH rating and crank amps.
Of course a HO alternator would definately benefit. But make sure you get one with a high output at idle. Even though your alternator at stock puts out 110A max, thats at a higher RPM. Who really drives at high rpm's all day or wants to rev the engine to listen to music? A HO alternator will still not benefit that much without doing the big 3 with 1/0 gauge. My alt can put out 200A at idle, or 270 at around 2k-2.5k rpm's. Even with that, there is still voltage drop, and I have a optima under the hood, shuriken bt-100 in the trunk, all 1/0 wire, including big 3, and several grounding points. The drop in voltage is only one tenth or two tenths of a volt, but still present. Doesn't do anything like dim lights since it's so small, but still. Now if I rev the engine, no voltage drop at all.
I would save the alternator for last unless you have several hundred dollars for a decent alternator. In all, do the big 3 and get a new starting battery before you get a HO alternator.