Not sure how much research you have done on this - but your options are basically limited by time and funds. Not a whole lot you can do with the stock 1ZZ-FE engine to make it more powerful, already been optimized for good life and fuel economy. To bump up the power - that will not been very cheap (assuming you want to do it right and reliably). Bolt-ons will help with making the car sound and look better - but don't count on them for any additional power. Even with the best I/H/E setups - might see maybe 5-10 HP gain tops. Not sure what you mean by lightweight belts - did you mean lightweight pulleys? Those will help with engine response, but not generally bump up power. In worst cases - you could damage the engine, as those pulleys will replace dynamically balanced OEM pulleys. This will be lessened if you buy quality pulleys from a reputable manufacturer - but to have a shop mill one out of billet, that's hit of miss.
Engine swaps are a possibility - which one to get depends on what you want from the car and what your budget looks like. They can be pretty mild to pretty wild - you can pretty much do anything you want, with enough time and money. I've seen swaps with older 4AGE/4AGZE and 3SGTE engines - there are few with the Supra 2JZGET/2JZGTTE straight-6 turbo/twin turbo engines stuffed in them. V6 from corporate Toyota offerings are also pretty popular. Another owner on a forum stuffed the V8 from a wrecked Tundra into an 8th gen chassis. So pretty much whatever you can think of, is possible. Cost wise - example - 3SGTE + AWD drivetrain into a 9th gen Corolla chassis - cost a company $40K to swap in. Sure it would be a riot to drive - but for $40K - I could buy an older C5 vette - hand it over to Lingenfelter end up with a Vette that will walk the new Supercharged ZR1 Vette at the drag strip and still have money left over to buy another car.
Ones that are still "doable" with minimal fabrication - 2ZZ-GE swap from the 7th gen Celica, 9th gen Corolla XRS, or 1st gen Matrix XRS/Vibe GT are a possibility. Have between 164-180HP right from the start - still mates up to the existing mounts, also can be fitted the existing transaxle (though it would nullify the power band of the 2ZZ-GE). Little more work needed on the electrical side - but that would be significantly less work that dropping in a completely different engine and gutting/fabrication of the chassis to fit the requisite drivetrain.
Before you get too far ahead - have to see if the swap is even legal in your area. Some places, you can make the swap but cannot title the car for highway use - be completely restricted to track use only. Not sure if you are looking for a trailer queen - but a legal swap is something to identify.
Have you considered boost or other power adders? Turbocharging / supercharging the existing engine - running nitrous? These are significantly less complicated / generally less expensive than swaps.
For modifications - stick with things that cater to the car's strengths or known weaknesses. It is an economy car - so any modifications to the suspension and brakes/tire/wheel will make for a very pronounced improvement. For relatively low cost - you can completely transform the feel and ride of the car. Car is already pretty lightweight - so thinking of reducing weight even further will help performance in the end. This will be counter to the sound system that you put in, as those tend to do better with heavier parts (dynamat or similar matting, heavy speaker enclosures, heavy duty speaker baskets, heavier magnets, thicker gauge wiring, amps, caps, possible battery upgrade, hi-po alternator, etc.) All depends on how extreme you go with the sound system. If a modest upgrade - I wouldn't worry about it. If you are looking for something for SPL or dB Drag competitions - that is a completely different story.
