Nice start there, Wurf, for OP.
My two bits is that an aftermarket intake, just about any reputable brand, either SRI or CAI (which I prefer) will dramatically improve performance in this car. I don't know why Mazda put such a restrictive intake on this car. The major car magazines, both Car & Driver and Road & Track dynoed this car before and after installing the Mazdaspeed branded CAI (made by AEM) and saw about 20 wheel horsepower and 20 ft. pounds of torque gains. That is tremendous for such a simple mod. SRI's show similar gains and I won't bog us down in the never-ending debate about which is better. Either will produce profound power increase over the stock intake.
And don't, I repeat, don't buy an aftermarket BOV and try to run it vent to atmosphere on this car. This engine is controlled by an electronic brain that depends on a signal from a certain sensor, the mass air flow sensor (MAF). That sensor is located between your air filter and the inlet on your turbo. It measures the mass of air crossing its sensor and uses that air mass value to calculate how much fuel to tell the injectors to squirt into the engine under incredibly high pressure (over 1600 psi). The stock bypass valve (BPV) works extremely well, but if you have one of the rare ones that develops a leak, for gosh sakes get a replacement that is set up to recirculate that already measured (metered) air back into the intake. If you vent to air, you are highly likely to have a rough running engine that stumbles at low rpm, that runs excessively rich and is not much fun to drive, at a minimum. The engine expects that air to go into the cylinders through the intake valves to match the fuel being shot from the injectors. When that air goes to atmosphere, the engine is being deprived on air the computer says it should be getting.
And don't go spending a lot of money on a catback exhaust expecting a power gain. The stock exhaust flows extremely well and little to no power gains can be made by changing it out. There are restrictions on the exhaust side, but they are upstream in the badly designed downpipe and two (yes two) low flow catalytic converters. Cat back exhaust (CBE) will only produce a change in sound, and such exhausts are expensive.
Intake and a rear motor mount, like Wurf said, ought to be your beginning point.
A good way to tell what mods are working and how well the work is look at the mods list on the signature line of some of the more frequent and respected posters here and use the search function to do some reading. About that: our search function is not very good IMHO. A good way to search this site is to use Google instead. Just begin your Google search terms with "mazdas247" (without the quotes) followed by the search terms. It will take you to this site and then conduct the search.
Enjoy you Speed 3. You're going to like it a lot. Welcome to the forum.