Horsepower doesn't mean doodly. Storage Capacity (size of tank in gallons) is good because it means longer times between run cycles, but it's still not the most important spec.
CFM - Cubic Feet Per Minute - is THE critical specification for any compressor. It's telling you how much air the compressor can push at a set pressure (usually 125 PSI) at full load. If a compressor is rated at 15 CFM @ 125psi, that means it will put out a constant 15 CFM at that pressure with the compressor running full time. More CFM is always better, especially if you're running a blasting cabinet or a high volume air tool like a grinder or sander.
To belabor the point a bit, a 5hp motor turning a compressor head that only puts out 9 CFM is not as good a deal as a 3hp motor turning a compressor head that puts out 12 CFM, all other things being equal.
All you NEED is a regulator and a quick connect. Most compressors will come with these, and you can hook a hose directly to the compressor and you're good to go.
Piping and valves are nice to give you the option of plumbing outlets where they're the most convenient instead of having to drag hoses all over the place. I have one at my workbench, one in my woodworking shop and one by the overhead door with a hose reel.
Filters and water traps are nice if you're running a blasting cabinet or painting. In line oilers are nice if you do a LOT of work with air tools. DO NOT run an oiler on a line that will be used to run a paint gun, you don't want any oil residue in your paint.