Re greasing; most non-bicycle applications use grease nipples (which the Yanks call 'zerk fittings' for some reason) but bicycles often use a simple ball fitting or just a plain hole e.g. if it is a DIY conversion. you need a pointy nozzle for these applications. Proper grease nipples and ball fittings have a check valve in them that doesn't leak out backwards even under high pressure. Well, that is the idea, anyway. Fortunately bicycle parts don't need high pressure grease application.
Dualco make some small bulk-fill grease guns which come with a pointed nozzle; for occasional use on bikes these are fine, but the nozzle isn't very long so access is poor in some cases. Theses guns appear to lack any kind of bleed valves.
R.S. sell a small-ish pistol action bulk-fill grease gun which has a single-stage bleed action and comes with a fitting for conventional grease nipples. It is a pretty good little gun for the money but you will need a pointed nozzle converter for many bike fittings and the single stage bleed is a bit of a pain, moreso on conventional nipples at high pressure than otherwise, admittedly.
There are many other grease guns out there, many of which accept grease cartridges rather than bulk fill. If you intend to buy grease in cartridges then maybe it makes sense to buy a gun of this sort. Bulk filling of small guns from 1lb tubs is fairly easy; you just take the spring pressure off, remove the reservoir from the gun, then push the reservoir down over the delivery plate in the tub (the one with a hole in it) until it is full. Stick it back on, spring back on, bleed it out and you are good to go.
Probably the 'gold standard' in small grease guns are those made by Wanner (now wanner-abnox). Like this;

These guns use bulk fill, have (usually) a two-stage bleed, and are 'last a lifetime' quality, with full spares backup. The one in the picture is an older model but the internal spare parts from the current 'MiniWanner' model fit perfectly in most cases. The delivery tube is mounted via a 1/8" BSP thread, so you can use other fittings easily enough. Once fitted with a pointy nozzle they are tops for doing bike work. The seals are usually good enough that you can use these guns with oil if you want, but I guess it risks leakage when the gun is stored.
The downside is that these guns are pricey; about £70 new these days, but if you are lucky you will find a used one for a lot less than that.
Note that with any conventional grease gun it is a very bad idea to leave the spring pressure on the reservoir; this causes the oil to separate from the soap in the grease, leak out, then leave the soap behind in the gun, usually in a solid lump after a few weeks.
Mad thought; I have wondered if you can make an inexpensive 'grease gun' by refilling a sealant cartridge with grease and using a small nozzle with it in a sealant gun. It wouldn't do high pressures, but they might be high enough for some bike greasing operations.
cheers