It all depends on the space under the hood. If it's not too tight, you can add a sequential turbo (you need drivability for this car, not top-end). You'd need a blow-off valve, a boost controller, new oil lines, all that intake piping, an optional intercooler, and a wastegate if the turbo doesn't come with one.
With extra boost comes the additional difficulty of firing off the stock ignition. MSD/Crane distributor-less ignitions are excellent - you'll need new coils, and maybe spark plugs. Fine-wire iridium spark plugs will let you fire easily, but they are costly. I'm no expert on Avalons, but the spark gap should stay below .05", depending on boost level.
Additionally, if boost levels are high, you will need a new fuel pressure regulator, and maybe a new fuel pump. Higher-flow fuel injectors will also be needed if the engine is constantly running lean.
To take advantage of all this, you need a modified ECU. Go to a shop that does this.
You'd then want a mandrel-bent exhaust to reduce backpressure (no matter what ANYBODY tells you, BACKPRESSURE IS NOT NEEDED AT ALL!!!), preferably with a diameter of over 2.5", and a straight-through muffler. Magnaflows are nice - and they don't sound like a wacked-out 2-stroke lawnmower. Go for the oval stainless steel ones.
Porting the areas where air flows is costly, and to many people, just isn't worth the price. However, a custom cold-air intake situated near *cough* a 'custom' hole in the fender well would do well to free up some asthmatic flow.
Contrary to what somebody in this thread said earlier, DO NOT disconnect the compressor. The Avalon is a family car and should stay that way (but you can make it faster. 🙂).
On the handling side of things, install bigger vented & slotted brake discs. Brembo kits are nice but costly. Switch your brake pads to EBC Greenstuff pads. Excellent stopping power with very little dust. Then you'd want to pick up lightweight forged aluminum wheels with low profile tires like BFGoodrich g-Force TA/KD[WS] (depending on weather), to reduce unsprung mass and to improve steering response. However, you should end up with the same outside diameter of the wheel, or else you'll have to replace the speedo and odometer. BTW, the thinner you go (sidewall height...) the rougher the ride will be.
Coilovers will let you lower the car for a lower CG, but too low and you will be classified as a ricer 🙂. These are totally optional.
Aerodynamics? You don't need any. Please don't put a wing on your beautiful car.