It does depend on your (and your parents'!) budget...

If you are looking for the cheapest possible economy class fares then I would not worry too much about the FFP and just make your travel on a purely transactional basis. That said, if you join either AA, BA or US you will be able to earn points on each of the others due to their oneworld membership.

If you have some discretionary spend and fancy travelling in business class but need to secure a good deal on the fare then I would recommend BA. Their upgrade-using-Avios deal is extremely good. It allows passengers to pay a premium economy fare of about USD 1,000-USD 1,500 and travel in business class for 10k Avios each way. You might need to do some work to secure reward seats but if you can get them this is a tremendous deal.

Four premium economy returns upgraded to business class on BA will earn you Silver in the Executive Club, equivalent to intermediate tier Sapphire in the oneworld hierarchy.

If you have the budget for business class or first class travel under regular fares then I suggest that you make your decision based on the product you like the look of the most. I can recommend BA as can many others but just as many will recommend another player.

Aside - travel around the UK

I believe that trains are the best way to get around the UK. You might want to sign up to East Coast's reward scheme, which gives points for every pound spent on rail travel (on any operator, not just East Coast) through their web site.