My fastest on a bike ( so far) has been 35 mph but 59 mph is totally mad. I think I'd be on the back of the tandem with my eyes shut!

On the flat , doing a 10 mile time trial against the clock I've done 10 miles in under half an hour which works out at an average of 20mph, but certainly don't come anywhere close to this on a normal training ride. That's around 14-15 mph, but as Veronica pointed out that can change according to the weather, my energy levels,etc.

Riding time trials is a very specific type of racing as it's just you against the clock over a set distance . Inverals can help build up the speed in your legs and get you used to the idea that it's gonna hurt, but some of it is down to technique - knowing just how hard you can push yourself, so you can ring every last ounce of energy out just as you cross the finish line, rather than getting a mile from the end and finding you've totally blown.

As for on a training ride, this will come with time as you improve. My average speed has been creeping up this year as I get stronger, but for me it's been a slow process. My advice would be to listen to your body and see how you feel. While you need to push youself to improve, you don't want to wear youself into a frazzel by trying to stick to a speed that's too fast for you. Just take it in small steps - afterall, increasing you average speed by even 2 mph over the duration of ,say,a 3 hour ride is a significant increase of effort.