Is your flying going to be exclusively BA? And in which classes?
If you make BA Executive Club Silver or Gold, you will get partner benefits across the oneworld alliance as well.
You won't earn points towards status on BA if you fly on discount economy fares - but subject to that you will earn points towards status on most oneworld flights as well as BA flights. (Most notable exception - no earning on AA from London to the US.)
If you want to use the BA lounges when not flying a premium class, the only way is to hold a BA Executive Club Silver or Gold card or a oneworld partner equivalent, plus the Qantas Club (Qantas paid lounge access scheme). BA is a partner in that scheme so if you're a QF Club member you can use BA lounges when flying BA.
The paid stuff that gets advertised in the glossy magazines generally won't get you into BA lounges, because BA doesn't participate in those schemes. However, it will usually get you into some other lounge - for example, Priority Pass will get you into KLM's Holideck at LHR T4.
If you do four return trips a year to the US in business class, you should easily qualify and requalify for BA Silver every year. If you need to pay for these trips yourself, then the cheapest way to do it from Europe is to buy a premium class ticket from Lisbon when BA have promotional fares originating there. You have to fly there to start the ticket, but they are very cheap when they're available.