First off, Jack's spot on about all the basics... which is really all you should worry about at level 19.
The only note I would make is that, while Zen is the currency that is purchased for 'real world' money, there is interaction between the currencies (most obviously, facilitated, by the Dilithium Exchange, which is currently running at the low 200s of Dil to 1 Zen). I would also argue that Cryptic's only ripping us off if you feel like the valuation of what you paid for is worth less than what you paid for it.
The fact of the matter is, you're going to change gear so much and pick up so many DOffs and BOffs on the way to the end of the game, that you don't want to make any big purchases yet.
Next, to your specific questions:
1. How to get Duty Officers (DOffs) required to complete certain Assignments - there's a lot there, but, simply, you can buy DOffs on the exchange and get them from missions and assignments. To build up your crew, three things are kind of mandatory. One is simply going to Starfleet Academy and going to the Personnel Officer in the little building past the pond (in the same booth as the cat-dude, but you gotta go right up to her on the other side) every couple of days. Another is to complete Colonization Cluster assignment chains (marked as little silver boxes with busts/heads in them on the map) and relocate refugees. This means you need to pick up every group of colonists and refugees you can as you look at assignments on your way through space. For a more in-depth take on DOffs:
http://www.reddit.com/r/sto/wiki/advanced_doffing
2. You accumulate large amounts of EC (energy credits) by various means... I mix up crafting (you'll get into that in a level or two and should do you research there, too), running missions (and selling junk loot to vendors), and, mostly, playing the Exchange. In the beginning, I'd suggest playing around with the sort settings and seeing if you can find some commodity or another that's stackable where the price-per-unit on a stack is far off from the price when sold individually. Then, break it up... it's tedious, but can get you a good war chest if you're diligent. While you're doing this, start charting out some higher-ticket items (lockbox ships, traits, high-end materials, etc.) and see what a high and a low price are. Buy low, sell high. Bang. Hit the lights and cue up the Donna Summer, it's party time (or, rather, a long grind and lots of risk-taking because the in-game economy is turbulent).
3. Lockboxes - someone else might be better to speak with about this (I know folks like Kenthend love opening them and I love their scraps, lol...), but lockbox keys are very pricey and lockboxes are not. When you open one, you get lobi (the only guarantee, and not a lot per box, but the only consistent way to get lobi), as well as some other items... there's usually useful stuff in there, but it's a gamble. You might end up with stuff worth a lot less than a lockbox key or with a sick ship that everybody wants, in which case, you made it (still, don't waste your newfound fortune, share some of it with me!).
That said, there's no better time to start learning yourself on earning than right now. And, if you joint the STOA fleet, you'll get access to some graphs of stuff I've been tracking to give you a head start on that EC-making, as well as plenty of great tips in chat about time-sensitive ways to earn stuff in the game.
On that note, you'll want to track in-game events (like the Dilithium Weekend that just passed) and how they affect pricing... what commodities are limited-time only and sellable, how scarcity or abundance (say, when there's an event that gives a special reward for a particular mission that rewards Rubidium, the price of Rubidium plummets, as all the players going for the reward dump it in the exchange) affect value.
Dunno why I wrote a wall of text for three simple questions... I guess I felt I had to earn the title "expert," lol, but take it slow, don't buy much (if anything) at this point, and start watching prices... you'll get a feel for things by the time you need money (unlike me, who waited a really long time to get up on my in-game money).
I have sold a couple of things on the Exchange but need to get more in depth.