Go via LHR and you will have the best Avios options for sidetrips.
Lisbon, Athens and Istanbul are some of the cheaper (as in daily costs) cities in Europe right now. But they can be too hot and busy in July/August, as is most of Southern Europe..

If you're going July/August, I'd suggest North of the Alps - maybe Munich or Amsterdam. Amsterdam is surely the easiest introduction to Europe if you've never visited before, and a great city any time of year. Nice cycling round about. Munich has great lakes and parks and is not too far from mountains. Avios has some great hotels in these places.

In October it's hard to beat Madrid.