It's really easy to spend too much money when updating a bike. Simply buying new tyres for that bike MAY be overdoing it.
One route is to slowly buy good components and upgrade it as you go. This route treats the frame as just another component, provides the greatest chance of spending too much and leaves you with one bike.

A second route is to buy a cheap(ish) bike already built and use that to learn about fixed gear - there's a really good chance you'll find the reality doesn't match your expectations but you WILL learn where you fit in and enable you to plan your next, expensive bike. A bike that's too poor to ride properly will not give you this experience. As a bonus, your current bike will still be there for beater duties.

The third route is to spend up and buy a really good bike now which carries the danger of buying something you discover later isn't appropriate for your needs ( the number of low mileage roadies for sale speaks to this).

My suggestion is the second route, I've spent too much money pursuing the other two not to