Obviously, if you want to use the train a lot, you will want to use a rechargeable option. You didn't explicitely address this in your question, but even with AAA batteries, you can either use disposable batteries or rechargeable ones.
The cost of using disposable batteries can very fast become prohibitive, but rechargeable ones tend to cost more upfront (especially if you need a charger as well). Even there, you have several options, including "stay-charged" batteries which you may want to consider if you need the train to be available as soon as you want it instead of having to recharge the batteries prior to using it, which can take a long time especially if you don't have enough chargers for all batteries at the same time.
So the first thing is to consider if you're already using rechargeable batteries for other things - if you already have a good charger or two and enough AAA batteries, you'll of course want to start with that and judge if it's enough for you. If you don't have any chargers or batteries, then I'd recommend to start with the official LEGO battery right away, as the cost difference won't be that high. Of course, if you're somewhere in between (say you have chargers but no AAA batteries), you'll have to ponder which route is better.
Another factor which may influence you is the convenience. Using AAA batteries and chargers can be cumbersome (again, especially if you've not enough chargers to charge all 6 batteries at the same time), and even the action of swapping the batteries in the battery box can be annoying. The LEGO battery doesn't need to be removed, usually just removing the train roof is enough.
All in all, I prefer the official LEGO battery; note that you could spare some money on the charger if you manage to find one which is compatible (although it's officially not recommended).
As for the battery box itself, either you buy a motorized train and you get one with it; either you want to motorize another train and then it's probably worth it to buy the basic city train anyway to get all electrical parts rather than buy them separately (since you get motor, receiver, battery box, remote control, and tracks, it's usually a better deal even without considering the rest of the parts).