Cycling is in main a sport dominated by men and a lot of these men will tell you that you need to spend lots of money on a bike with a high spec and low weight. Don't listen to these people. Bicycles haven't really changed much in terms of mechanics since the chain driven models were first introduced in 1885. Sure, there are some amazing bikes that I'm sure if you have a few grand to spare are worth every penny and will shave precious seconds off your Time trial time. However, if you're looking for a nice sturdy, reliable, easy to maintain, comfortable ride which won't cost the earth then Ebay may be the place to go.
This little beauty is going for £99 at the moment, a bargain if you ask me.
Now here's where these vintage cycles will divide opinion. They may well need a lot of work but so long as the frame is sound there isn't anything too complicated that can go wrong. There are also a lot of websites out there to help you out if you need it, like the Park Tools blog, Evans will even teach you how to do if for free. When you learn these skills they stay with you for ever and will help in the future if the wheels do fall off.