As long as the moving parts (esp. wheel and bottom bracket bearings) are in good condition, spending any money to make your bike faster will only help by making your wallet lighter.

1. Make sure your fit is optimized on the bike - you could be limiting your power output if your saddle is not in the correct position, of if the fit of the bike is causing you fatigue.
2. When you need new tires, consider good quality ones that are lightweight and have a supple casing. Cheaper tires and tires with a flat protective layer generally sap more energy as you ride than better quality tires. This is a tradeoff you have to consider - the slowest and heaviest tires might be the most puncture resistant, while the lightest and fastest tires are more likely to suffer a puncture, which slows you down a lot more than the heavy slow tires.
3. Make sure everything is in good condition. For example, an improperly lubricated or very worn chain or drivetrain will take more energy to keep spinning.