This seems to be the most commonly asked question. For those who want to research and understand how everything actually works and what all the pluses and minuses are, these questions have all been addressed and a great deal of information is available by searching. Researching each individual component you put on your car is the recommended approach. Beyond that, you must make the final decision and take responsibility for everything you do to your car. This thread is intended to simply summarize the most commonly recommended options and to provide a brief explanation for the reasons.
For the most part, the answer is pretty much the same whoever you ask when discussing 02-07 WRXs and STis. I am not sure what is up with later years, maybe someone else can add any differences.
1) Uppipe: Only required for '02-'05 WRX (no cat in other years/models stock). This is to replace a cat that we have all seen pictures of becoming detached and entering the turbo (bad). Do this first because everything else increases that chance. Very generally 10HP+ and 500 RPM sooner spool is the bonus. PITA to install but cheap to buy.
Attn '02-'05 WRX Owners: The Importance of the Uppipe Mod
Uppipe, Downpipe, Muffler INSTALL GUIDE!
2) Downpipe (bellmouth or divorced wastegate): All years/models. Replaces two restrictive cats. A highflow cat is recommended on 2.5l models to restrict the possibility of boost creep and is my preference on 2.0l. MUST be tuned or used with a stage II map (except '02-'03 some say but do it anyway). Gains are substantial with a tune. Read on boost creep if you have a 2.5l.
3) Engine Management: Opensource reflash, Cobb accessport, or professional tune. Increases your boost safely with fueling and timing adjustments etc. etc. and allows you to run a DP safely. Also very beneficial without a DP (stage I).
4) Catback: This is only for your sound preference. Gains are minimal for the cost. Don't fall into the trap of getting something too loud unless you are sure you like that. Match to the DP for connection - options are 3" flat flange or ~2.5" doughnut (stock)
5) Intercooler hoses: Not that often recommended here but is a great mod for the $. Better flow and maybe reduced temps. - gain is 5hp or 8hp according to the famous author depending where you read.
-1) MBC - Only run these because your tuner recommended. Not a safe way to up boost except very marginally on '02-'03WRX and maybe '04-'05 STi without EM and correct tuning. Engine management is a FAR better option to get more boost and HP on all models. Does not account for extra fueling etc. needed. Issues without tuning include partial throttle full boost, causing bad lean conditions and horrible drivability.
-2) Intake - not necessary yet. May skew your MAF causing you to run lean. If you want one, make sure that it is certified to not mess up your MAF scaling and make sure no one has run into problems or better yet get it tuned. May be a consideration for stage 2 STi.
-3) BOV or Blow off valve: just a noisemaker and not really ideal for your car.
-4) Turbo Timer: Not needed. The turbo is water cooled even after shut-down.
At this point, your biggest gain/$ will be to do your turbo and supporting mods (intercooler, fuel pump, injectors, tune). The most common standard recommendations are 16G EVOIII for 2.0l or 8cm 18g for 2.5l WRX (you are really slacking if you don't research this). Consider an even larger turbo to upgrade the STi.
A variety of other things can be done still though to improve flow, cool the intake charge etc. However, manual boost controllers, intakes and Blow foo valves are not on the list for reasons.
6) Intercooler - STi is already a little better than WRX and can be used to upgrade WRX - small gain little bit cooler charge. High quality big TMIC gives less pressure drop (at least 2 PSI), faster response, less chance of det. and can be tuned for ~15-20hp. Supporting mod for turbo upgrade.
This is pretty much of the top of my head so if you see an error I will correct it.