Originally Posted by DMF

That's called a "six-speed". We count the rear gears.

I'd be most concerned about the gearing. You will want a certain range for your best performance. If you can get that range then the number of speeds isn't that important. Nor is indexed shifting.

You'll eat up most of your budget with bars/shifters/saddle and tune-up parts. Also have a mechanic evaluate your chain wear. You may need to buy a chain and cassette. And front chainrings can wear out too.