You only need a volt/amp metre.
Measure the voltage on the battery with the engine off. It should be 12.6v. If it isn't then the battery needs a charge. Charge it up for a few hours. Let it calm down a bit.. the voltage should stabilise at 12.6v.
Measure the voltage again the next day. If it's dropped to below 12.6v overnight, then either it's not holding a charge or something is draining it.
Charge it again but this time leave the battery disconnected. If the voltage still drops, then the battery is knacked. If it doesn't, then something is draining it.. and you'll have to isolate each electrical circuit, one at a time, until you find out what that is.
Radio is usually a good candidate... as is alarm and internal lighting (including in the boot). Take the fuses out until your current draw drops to zero.
To test the alternator... start the engine with the lights on. Rev the engine a bit and get someone to tell you if the headlamps get brighter. If they do, then the regulator could be MIA. Measure the voltage across the battery with the engine running. You should have 14.5V and that shouldn't increase or decrease with blipping the throttle.
If your voltage doesn't increase unless you give it in excess of 3000 rpm say, then the alternator or a component is knacked.
My guess is that the battery became moody at not being used and became sulphated. At low charge (sitting around for weeks) the battery acid can attack the lead plates, forming lead suplhate... it's a white powder that is conductive. It settles in the bottom of the battery cells and eventually rises to the point where it touches the plates, effectively shorting them out and reducing the battery's cranking ability. In some cases, the lead plate gets so thin (eaten away) that it fractures and falls into the battery, shorting out the plate each side.
If you have a delinquent cell, your battery voltage will never rise higher that 12v in this situation, since you're left with 5 good cells (and whatever is left in the knackered one - maybe 11.x volts altogether) - enough to turn the beast over .. but not for very long.
New battery is the only fix.
Cheaper and easier than an alternator problem.. so could be worse.