Have I got enough pins in my Uno or should I use another board...?

Ron, why do you need so many more pins? Future expansion? Your current requirements require only 15 and Uno has 20. If cost is a factor, a Pro Mini, Pro Micro or Nano 3 would also suit. Smaller too. And all those have the same chip and about 100 times more speed and memory than you will need for what you have described so far.

Cheapest place to buy from in UK is from China on eBay, same as everywhere else. Cheapest UK sources are, again, eBay. Megas from aroubd £12 versus £30+ in most other places.