Sharon
Most fountain pen kits use the 'international standard short' cartridge and/or converter. These are widely available, although the suburban big-box stationery stores (Office Max, Staples, etc) don't always carry them. Fortunately, on-line suppliers such as Goulet Pens, Jet Pens, etc., do stock them. In addition to kit pens, these also fit a number of name-brand pens.
By the way, specialty stationery stores in large cities (especially in Europe) almost always carry fountain pen ink cartridges.
There is also an 'international standard long' cartridge, but they are harder to find. Any pen that is designed to use the 'international standard long' cartridge can also use the international standard short cartridge. And a neat trick is that pens that can hold the long cartridge can often hold two short cartridges - one actually plugged into the pen and in use, and the other stored in reverse position inside the barrel. That means that it is possible to hide a spare cartridge inside the pen - something that is handy when traveling.
As to brands - any brand that also sells bottled ink is fine. I bought a new pen when I was in France a couple of moths ago, and picked up a box of Pelikan cartridges at a stationery shop in Arles. Other good brands are J. Herbin, Diamine, Faber-Kastell, Visconti, Kaweco and Private Reserve. Just avoid the unbranded 'bag of cartridges' from China option on e-Bay.
Higher-end non-kit pens sometimes use a proprietary cartridge design, for example, Cross, MontBlanc, Parker, Lamy, etc. And not surprisingly, those refills are also more expensive (that's the Gillette razor marketing philosophy in practice). In is generally possible to purchase pumps (converters) for these pens as well. Big-box stationery stores may carry a limited selection of proprietary cartridges, but usually only for the brands of pens they sell.
I use converters in my fountain pens. Converters (aka pumps) can easily be refilled from bottles of ink (and ink in bottles is going to be much less expensive than ink in a specialty, throwaway cartridge). Converters almost always hold much more ink than disposable cartridges. And the selection of inks in cartridges is limited, while there is an unbelievable choice of inks available in bottles - with a very wide price range.
Cartridges can easily be refilled using a blunt syringe (sold for refilling the cartridges for ink-jet printers). However, its difficult to reseal the end of a cartridge, so this is usually only practical for refilling cartridges at home or in the office, and then immediately installing the cartridge in a pen. Eventually, recycled cartridges do wear out - repeatedly removing and then reinstalling them causes wear on the plastic nipple that connects to the pen, but my experience is that this takes years to occur and is usually not a serious problem.