(Updated, now I've actually tried it!)
When first introduced, domestic guttering was made of iron or other bulky metal. Iron rusts when damp, so one might ask why use it. However, this provides a wonderful opportunity for the handy DIYer - replacing iron gutters is an exciting, high-risk activity best undertaken with one or more friends.
Guttering tends to run above head-height. That's so the unsightly stuff is out of the way. If you're taking down or putting up guttering, you'll need a good, stable platform to work from. (Unless the idea of large lumps of iron dropping on your head sounds more fun.) Getting your friends to help at this point is a good idea.
Modern gutters are plastic, making installing the new system less exciting...
If you're going to put guttering back up (a good idea - it was probably there for a reason), you've some choices to make - style/colour and size. Basically, you need at least the same water capacity as the old system - if it's a new system, you can calculate the capacity required - check with your supplier. Remember when measuring up to round up to the size of the gutter and pipe lengths you'll need and to avoid joining lengths.
You'll want the water to run towards your outlets. Allow at least 5mm in 3m drop towards each outlet - use a spirit level to get the level, then adjust. Offer up an offcut of gutter and clip to see what height you need to start at. Check the line as you go - old houses can be at odd angles, so you may be getting too close to the tiles if you rely on a spirit level. Make sure you leave enough room to actually fix the clip!
Once you've got your line, fix your clips. Use non-rusting screws! Keep clips about 100mm from corners and ends to allow for these fittings. Check the fittings to see where and whether they want to be clipped. Clip at least once a metre along the runs.
Fix up an outlet. If you need to, fix up the corner you'll run to. Now measure up the gutter (leaving about 5mm expansion into fittings and allow 50mm beyond the fascia at an end), cut square, deburr and fit. Repeat until all your gutter is in place.
Fitting the drainpipe is slightly less straightforward. You probably got away with just a screwdriver for the gutter clips. You're going to need a power drill and wallplugs, too, for this.
First, and most obvious, site your downpipes vertically above your water discharge points (gulley, drain). Now, take a pair of 112°-angled connectors and join them with just enough pipe to connect from the gutter outlet to the downpipe. This should be neatly against a wall. Mark where on the downpipe needs to be cut (keeping the shoe 50mm clear of the ground). Cut, clip, fit.