Personally, I ignore the stuff in their zen store. Some of it does give anyone who buys it an advantage, but most of STO is PVE, so unless you're doing PVP it doesn't matter too much. Some of it expands your inventory space or bank space or the like, but that's stuff that you don't necessarily need.
You get can get zen from a zen <-> dilithium exchange where other people sell zen for dilithium. You get dilithium from a bunch of different in-game missions. I've never gotten a meaningful amount from them because I mostly ignore those missions, but you could farm a substantial amount on a daily basis if you wanted to waste the time doing the daily missions which give you dilithium. However, some vendors in-game sell items for dilithium, some of which are needed for certain activities like crafting (which is called research and development).
There are also lockboxes that are randomly dropped by enemies, to open them you need to buy a key - like TF2's lockboxes, only it's much rarer to get the rare spaceship in them than to get something useful out of a TF2 lockbox. Some people waste a ton of money opening them, so they can be sold for a small amount of energy credits on the exchange (ones which aren't being dropped at the time sell for more). (I consider opening lockboxes to be equivalent to gambling away your real money, and you're really just wasting it unless you like the idea of paying several hundred dollars for a spaceship)
That said, most things that are in lockboxes or sold for dilithium are re-sold on the exchange, for energy credits (the standard currency which you get by selling things), and if the price is within your reach you can buy them on the exchange. You can also list items you've found on the exchange where other players can buy them, instead of trying to sell them to a vendor or recycling them. Usually you get more money on the exchange, but some kinds of items don't sell (such as most hand weapons). Others sell for many times what is listed when you look at the item (so you need to do a search on the exchange to see what price the market is currently at for that item). Neverwinter uses a different exchange/market system, which is auction based (I assume they haven't changed it since I last played a week or two ago).)
Side note: Energy credit balance is normally capped at 10 million ec unless you have subscribed or something. It's possible to raise it (spending (500?) zen to do so) to 1 billion. Some items - such as the ships from lockboxes - sell for more than 10 million ec on the exchange. It is not difficult to eventually reach that cap by selling things on the exchange that you get just through playing STO normally, and to then buy purple equipment for your ship. (There are also STFs, though I have not done any)