This site is probably one of your better places to research dive destinations. Anyplace there's a decent amount of diving, people will have already written about it (ad naseum.) Search for Aruba or wherever and just read a while. You'll get a general idea what the diving routine will be like there, and which shops people recommend or not. You'll also learn stuff like what hotels you might want to stay in, if you haven't already booked one. (Some places it doesn't matter too much where you stay, because you won't be diving directly from the hotel anyway or the dive op will pick you up wherever. Other places it matters more, either because it makes sense to choose a dedicated dive resort with facilities right there, or because things are spread out and it's easier to get to the diving from certain locations. I think Aruba is more in the first category, but I haven't been there in ages and didn't dive there.) Once you've read a while you'll probably start narrowing it down to a few places and dive ops that sound good to you, and maybe you'll have some more specific questions to ask.
Generic travel sites can be useful for general hotel info, but aren't too useful for diving information. The best "dive" resorts may not appear on generic travel sites at all, or may be lower ranked because they were dinged by non-divers who didn't enjoy the smell of neoprene in the morning. If there is any talk of diving it's often from casual divers with different criteria than more frequent divers who tend to post their reviews on boards like this one.
Booking diving may seem intimidating if you've never done it and have no idea, but it's pretty simple. It's much like booking any other activity. You contact the shop you've decided you want to dive with, tell them when you want to dive, and see if they have space when you want to go. Many will require deposits. Often dive shops will also offer package deals with some convienient hotels they recommend and will book the hotel for you. Some will also offer packages with a vehicle rental included, if it's common where you're going.
You'll often see the term "2 tank dive" which is 2 dives done on one boat trip. A lot of places do 2 tank dives in the morning and 1 tank dives in the afternoon, but it varies. Depends on what people usually want and how far it is to the dive sites. You'll also sometimes see 3 tank trips which tend to be all day affairs to sites further away. Most places have package deals where it's a little cheaper the more days you dive with them. If you're going someplace with shore diving, you can often buy a package for tanks & weights only, or if you're buying a boat package they will usually throw in shore diving tanks for free. So you decide, you want to do the 2 tank trip every morning and sit on the beach in the afternoon, you also want to dive some afternoons, you want to dive every other morning, you want to dive as much as possible, whatever. Sometimes you can get away with not booking everything up front and add on later as the mood strikes you, but some ops are popular and will fill up, so that's something you need to ask. It seems divers on their first trip often don't think they want to dive all day or every day. Once they get hooked, they are figuring out how to fit in as many dives as possible.
There are as said many dive travel agents. They can be convienient, though for your average Carribean trip it's really just as easy and the same price to book things yourself. They start to be more useful and can possibly save you money on airfare if you're planning something more exotic. Often their websites will also provide some good info on what is available and commonly booked by divers. But any one dive travel agent will not offer all destinations, or all options in a destination, they tend to specialize and have places they work with. And if you use one you still want to do your own research to be sure the trip will suit you.
A regular travel agent will generally have no clue about diving and will recommend the same heavily promoted hotels and package deals they send everyone to. Best case you may get a good deal and an ok choice but you'll miss much better options for divers. Worst case it may be a nice place but a lousy choice to base yourself for diving.
Trips planned by a local dive shop can be an easy intro to dive travel, especially when it's someplace they've been before and know well. Again. do some research of your own to make sure it's a trip that will suit you.