Order of Precedence

Depends it's more of an interactive thing than something that you can nail down to each component and rate or order. With a killer pre but no or a crappy power section, well, you're outta luck. With a killer pre and power section but no speakers to send it though, well, again, you're out of luck.
for me
1) Pre/power/speakers
2) "Box" as you put it, though I'd count that in speakers personally
3) Manufacturer

 

The preamp and cabinet will set the tone of the amplifier. A good clean SS or tube amplifier will not add much color. A tube amplifier section that can be pushed will add some character, but most is done in the preamp.

The speaker cabinet for quality will have multiple quality brand name speakers, in a plywood cabinet of the correct size for the driver chosen.

For instance I have a bunch of the new plywood AMPEG SVT-810e cabs and a couple SVT215e 2-15" cabs. Both have nice punch and tone.

In my system those plus a quality preamp that has the tone I desire (SVTCL) and four SVP1600 power amplifiers makes up my current system.

Just one amp and one 810 cab would do the same thing, just smaller.

I give the cab and preamp #1 as a tie. The AMP is clean and the bass adds the rest of the character and is as important as the frist two.

BOB

 

pre-amp - whatever you need to shape your tone, plus maybe some basic functionality for ringing out whatever room you are playing in.
to my ears, tubes overdrive into a nice fuzzy saturation. silicone tends to be a bit more metallic and edgy sounding.

power-amp/speaker area - balance of wattage and speaker area to achieve your volume requirement, plus a little headroom.
tubes require a bit more TLC, but can add a natural compression. solid-state has a few different types, some are beefy and powerful, some are super light. i've heard people complain that the digital amps sound sterile, while others bless them for instant power.

speaker-type - again whatever you need to shape your tone. bigger cones aren't as articulate in the high-end response, and they may also be slower to respond. smaller cones are faster and can articulate better. depending on cabinet, any driver, even the best, can have a boomy or nasal character to it. look for a cabinet with thick sidewalls, strong bracing and internal padding. A solid box will not add any overtones of its own. Only the speakers/ports/baffles will be creating the sound.
look for a grille that doesn't vibrate.
neodymium vs ceramic/ferrous - weight is the biggest issue. I cant tell the difference aurally, though some people report huge differences. i'll leave elaboration to them.

Depending on where you practice/play, you might also want to find a set of gear (rig) that is capable of creating the tone you prefer at low and high volumes. e.g. you may achieve a great sound with the gain cranked, but that can be a hindrance when practicing at 2am. in that case you might want to look for something with a master volume as well.

The more components, connections and cables you add, the more you introduce potential sources of noise.

I would suggest if you're starting out, to get a combo. Pre-amp, power amp, speaker all in the same cabinet. If you need some more serious fair, a head (pre-amp/power amp) and a large cabinet arrangement should fit that need. Unless you're going on tour or have a fair amount of disposable income or really like tweaking things, you probably won't need a rack with separate effects, pre-amps, power amps and crossovers. Keep it simple is a decent rule.

Summarily- head to a music shop, play through everything, pick what you like that's also practical (i.e. not the amazing 810 that wont fit in your bedroom).

 

the less you worry about stuff like this at the point you're at, the better off you'll be. Play different amps until you find a handful you like, then make a decision.

i will, say, though, that the speaker cab makes or breaks an amp rig. if you have to cheap out anywhere, cheap out on anything but the cab.

 

Another big +1 to Jimmy, whatever you do, don't skimp on your speakers. If there is one thing that deserves to be in the #1 spot here, it's speakers. A good amp is important, but without good speakers, it's just about useless.