Start with credit cards for points
I'm in a similar position. Here's some background. I'm based in DFW and mostly fly American. I don't fly much, except for flights to visit family and occasionally for work, but I'd like to do more international travel for vacations. For example, last year my wife and I went on a vacation to Tokyo in coach using AA miles and stayed at the Mandarin Oriental and loved it (except for the flight). Last year, I had close to 20,000 elite qualifying miles (EQM) on AA, which was the most I've had in several years. So, without some mile runs or more vacation travel, I'm not getting elite status.
I've never flown first class or business class on international flights, but I occasionally buy the domestic first class upgrades at the kiosk. Last year, I got an American Express Platinum Card, mainly to get access to the Admirals Club (AC) (and I've also used it for the Delta Crown Room). Having access to the AC makes flying much more fun (for example, at the Tokyo AC and at the several DFW AC's) and it has come in handy when I've had layovers for work. If the club access is worth something to you, it is worth considering an AMEX Platinum. If you use the fine hotels and resorts benefits, it can help make the annual fee pay for itself, but it is $450, so you have to figure out how often you'll use it.
For the past few weeks, I've been reading FT threads trying to figure out how all this works. I'm sold on getting more rewards credit cards to accumulate more miles to get the first class/business class tickets. The recent BA Chase card (100,000 miles) is a great deal, but you can also get other credit cards with 25k to 50k free miles per card. You have to manage your credit score and be careful doing this, but for me, it is the easiest and cheapest way to accumulate miles for a big trip once every year or two. So, if you don't have much money to budget, I would just accumulate credit card miles until you have enough for the occasional big trip.
If you decide to travel more for vacation (for example, more short weekend domestic and short international breaks), then the AA executive platinum (EXP) (or another airline elite status) is doable. After the first year,you get lots of domestic upgrades to first class and four upgrades to business class on international flights, plus enough redeemable award miles (RDM) for some free flights. If you're careful, you don't have to do mile runs, but you have to travel on paid flights 100,000 miles per year (unless there is a double elite qualifying mile (EQM) promotion), which is a significant commitment. So it all comes down to your time and budget. I'm still thinking about this option, since it involves a big long-term time and money commitment, but the more I read about all this, the more fun it sounds.
Hope this helps.