I removed the old paint on my bike with a combination of coarse wet-and-dry paper, used wet, and a gas blowtorch for hard to reach areas. You don't need to remove all the old paint as long as the surface is smoothed into the bare metal. Hang the stripped bike up by the bottom bracket or headtube so you can get to all sides. Make sure to get all over the bike, first with a good coating of primer, followed by the colour coats. You can do fine detailing with a modelling paint brush and enamel paint (such as painting lugged joints in a contrasting colour). It's worth giving the finished paint job a few coats of clear to give it a bit more scratch resistance. Leave the finished bike for a few days before reassembling, to ensure the paint is fully cured and so prevent any clamp-on fittings from gouging the paint surface.
Of course, if you want to paint the bike in a solid colour it may be worth getting a professional powder coat, which will be more hardwearing and probably better looking than a DIY job.