I hear of more manuals then tips tbh.
Also, you say budget. You'll really need to explore exactly what you mean by a budget. If you have one, consider it already broken. Cars have an awful habbit and making you "need" the next thing on your list more than "needing" to stick to your budget.
Good advice previously. All worth thinking about, but the real question comes down to exactly how much you are willing to spend. You could very easily spend a few hundred quid and a few hours. Making it lighter is free unless you start going down the FG/CF route on boot and bonnet. A VERY hefty service on the engine and clean all the bits. It's time consuming but A LOT cheaper than a new engine. Also, buying an engine from a breakers could add to your troubles.
A manual is going to be a must for track use so bear that in mind. That said, you could still have a lot of fun in your tip but ultimately a manual is best.
There's no point in making your car faster until you know what kind of driver you are. Track and road are very different. More power could make you a worse driver so stick with the current engine for now would be my suggestion. Improve handling and safety above all and invest in a good helmet. Even a little knock could seriously hurt.
All in all, think about all your options and weigh up exactly what you want against how much you're prepared to spend.
Happy modifying