It depends on what your goals are for TATL tickets: Do you need just 1 ticket at a time (relatively easy) or do you need to redeem for a spouse or family as well (needs a lot more points going to one airline if you all want to fly together)? Are you looking to redeem for economy or premium cabin? The bottom lines is this: can you put enough spend on the CSP so that you will have enough (in bonus+spend points) to redeem for the ticket(s) you want in a reasonable timeframe? (BTW, you should consider opening a Freedom card as well, if you're going to go the route of applying for Chase cards first.)

Unfortunately, the only airline that allows transfers from all three major CC points programs is Singapore. You can get to Europe on Singapore (JFK-FRA), or you can book Star Alliance partner flights via Singapore - although they impose fuel surcharges. UA doesn't do that, so a reasonable strategy if you just need 1 ticket at a time would be to build up enough UR points for a ticket on UA or a partner, then switch spend to other cards.

In my case, I'm aiming for 3 business or first class tickets to Australia, which will require maximizing UR, MR and TY sign-up bonus points, maximizing category bonuses, buying a lot of crap over the next couple of years (I don't do MS) and then redeeming through Singapore (ideally, FOR Singapore tickets, as they have an excellent product).