It is a third world country, so expect lots of litter and poor people (especially but not exclusively illegal migrants from Haiti). You will encounter beggars and street vendors, but almost never are they aggressive. I would not worry about that and would be more concerned with road safety: Dominicans have a very relaxed attitude towards regulations and don't worry themselves about traffic signals, staying in defined lanes, etc. It is not that they are 'worse' drivers than us, they just drive 'differently'. Be especially careful of motorcycles, particularly at night (many do not have lights). Keep your head on a swivel!
There are numerous house burglaries and you will see plenty of security guards, typically armed with short-barrelled shotguns without stocks. This seems to be more of a deterrent than a necessity, and there are not a huge amount of violent crimes (less than in most US cities). But like Onward says, definitely use common sense: don't wear expensive clothing or jewellery, don't go out after dark, don't go wandering around in areas you don't know about, don't carry large amounts of cash on your person.
There is little if any genuine public transit, other than the metro in Santo Domingo. Private buses and 'taxis' (beat-up old Toyota Corollas plying fixed routes for fixed amounts, typically crammed with six or seven people) perform the same function, but are effectively unavailable to you unless you are confident and pretty fluent in Spanish. It would be best to arrange transportation through your hotel.
If you go to an all-inclusive resort, you will have no exposure to any of the above issues. You will see only beautiful beaches, excellent cuisine that is probably better than what you're used to back home, and accommodations that range from good to truly luxurious. Virtually all of the staff will speak fluent or near-fluent English.
There is a lot more to the DR than beaches: hiking and horse riding in the mountains are particularly worthwhile, and the Zona Colonial in Santo Domingo is worth seeing (North America's oldest city!). However, an all-inclusive vacation at one of the resorts would be a stress-free introduction.
While there are a few exceptions (like everywhere else in the world), most Dominicans are very friendly and helpful: which in my experience is generally true of the Caribbean. They are curious about strangers, so their manner can seem forward if you're not used to it. Just remember that you don't have to answer questions if you don't want to.
The country has a lot to offer, and I encourage you to give it a try. There are plenty of tourist books available. Another reference that is particularly insightful is Dominican Republic: Culture Smart! (2010), which you can buy via Amazon for less than $10.
