Cardiff Bay is a large area, and how convenient it is for social life depends very much upon where you are. If you're around the Scott Harbour or the Waterfront districts, then yes, social life is good with a huge range of restaurants, the Red Dragon Centre and the opera house. The Waterguard pub is to be especially reccomended.
On the other hand, if you were over more toward Atlantic Wharf, there's pretty apartments and impressive offices -- and NOTHING else. Not so much as an ATM! Never mind a convenience store.
The downside of the Bay generally is there's very few local shops. There's a big retail park the other side of the river, connected to the Bay by a major highway. It's got an Aldi and an Asda (which is basically Wal-Mart with a Union Jack). But if you just want to pop out for a pint of milk, you'll have a problem anywhere in the Bay.
I would say you'd be better off looking at the city centre, around the St David's Centre (currently being massively expanded into one of the biggest shopping malls in the UK) and the Marriott. This is where the Brewery Quarter is located, but I'd avoid the Brewery Quarter. It's a smallish development, arranged around a central square that echoes badly. The square is filled with restaurants and bars, which is cool except when the customers come staggering out, drunk as lords, just as you're trying to get to sleep. Behind the Brewery Quarter is Caroline Street, popularly known as "Salmonella Street" or "Syphilis Alley" -- draw your own conclusions. The other side of the Brewery Quarter is St Mary Street, which is one long party on Friday and Saturday nights. Again, cool -- except when you're trying to sleep in the Brewery Quarter a few feet above them.
Apartment blocks that I'd recommend close to the bars of St Mary Street and the shops of the St David's Centre would be Landmark Place, The Aspect and Altolusso. Altolusso, especially, has some very classy apartments and, being the tallest buildest in Cardiff, has some spectacular views from the higher floors. Try to get a north-facing apartment, since the south side of the city is, frankly, unimpressive.
As one who used to rent, I'd avoid the private listings. There's a big reason why privately listed lets are cheaper than the corporate listings, and it's got nothing to do with utilities. Frankly, many of the private listings suck. Many are aimed at students (Cardiff is a big college town), who tend not to be picky. A lot of others are aimed at immigrants, people dependent on social security, or low-income people who can't really afford anything better. A lot of them aren't even legal. You may have difficulty persuading the landlord to do anything efficiently beyond collect your rent. You will almost certainly never get your deposit back.
Be especially wary of anywhere that advertises itself as including a TV. TV's aren't generally included in furnished lettings, and most people have a TV of their own. If you don't have a TV of your own, it doesn't cost much to buy one. An included TV should tell you quite a bit about the people the landlord is used to getting.
Corporate apartments may be more expensive, but you can be reasonably sure that it's in good condition, comfortably furnished, that maintenance problems will be rectified relatively rapidly, and that your deposit will be returned when you move out.