What kind of ink should I use?
The answer to this question used to be simple: use an ink intended for use in a fountain pen not India ink or drawing ink.
That answer is no longer sufficient. There has been a huge increase in the number of inks on the market, many of them distinguished by intense and unusual colors, others by special features such as scent, gloss, rapid drying, and water resistance. Although sold as fountain pen inks, most are made by smaller companies that do not themselves manufacture fountain pens, and which therefore lack the overriding concern with safety and compatibility that prevailed back when most fountain pen ink was sold by fountain pen companies. And while ink from the big pen companies is generally safer, manufacturers no longer support discontinued products as they once did leading to new inks formulated for cartridge/converter pens and acrylic-barreled piston-fillers, with no concern for older pens, even of their own manufacture.
As a result, there has been a rash of reported problems. Perhaps the most common is premature failure of latex rubber ink sacs, with staining of barrels and caps a close second though there have also been more drastic issues, such as the actual disintegration of plastic (styrene) feeds.
So what is to be done? You must take the greatest care if your pen is made of celluloid or cellulose acetate, and it holds its ink in direct contact with the barrel or with other transparent parts, such as a see-through section.* With such pens, be very conservative in your choice of inks stick to brands such as Waterman, Parker, Sheaffer, and Pelikan, avoiding reds and purples and browns. If you use a highly pigmented ink in such a pen, staining is likely, and it may not be reversible.
Conversely, you can use pretty much any ink in a hard rubber eyedropper-filler (including safeties). Somewhere in the middle lie pens which hold their ink in a sac, cartridge, or converter, and which have hard rubber feeds and sections. Such pens will not be damaged directly by an aggressive ink, but they may experience premature failure of the sac, cartridge seal, or converter messy, but usually set right at moderate expense (silicone sacs will stand up to any ink available, but note that some sacs sold as silicone are actually PVC). Acrylic (Lucite) is very resistant to ink damage, making the Parker 51 a good choice for boutique inks (noting, however, that Vacumatic-fillers may still be vulnerable to premature diaphragm failure, while Aerometric fillers' transparent PVC sacs are vulnerable to staining).
PS You may be told to discard ink that is more than a year or two old. This is nonsense, as many pen collectors routinely use inks that are decades old. This is not to say that old inks cannot go bad; watch out for any sign of separation or sedimentation, or for off smells. If mold gets into a pen, it can be very difficult to eradicate, so any ink with slime or fuzzy growth should be thrown out, however new it is. Finally, there are a few special-purpose vintage inks that are strictly display items, or for use with dip pens only most notably, the Parker ink originally labelled "Parker 51" and later, "Superchrome".