Do you have the CF1 (alloy top) or later (plastic top) engine? The CF1 has "80" stamped in the middle of the temperature gauge, whereas the later engines had "90" in the middle of the gauge.
Note that "80" on the CF1 gauge is actually 90C..
Now then.... the thermostat should open at 86C (actual) and the fan comes on at 92C (actual). The fan cools the engine down to 90C (actual) before turning off. This sounds like what your car is doing (so I'm guessing you have a plastic top engine/90C gauge).
Because the fan stays on so long .. and then the tempearature climbs again, it suggests that the coolant is not being circulated quickly enough. Quicker circulation would see it more evenly cooled, rather than just a small volume of it. It could be a partially blocked radiatore, a faulty thermostat or possibly a duff water pump.
In terms of cheapness to test:
1) Radiatore: You should drain the coolant from the radiator and give it a severe flushing with cold water. Remove any gunk in there and try to clear anything that looks like a blockage.
If you have brown goo in the system, this is not good. Refill with clean water.. let the engine idle a bit.. drain the water and repeat... until the water comes out clean.
If you have just nice clear green or pink coolant in the rad'... it's not the rad'.
As an aside... don't tip old anti-freeze down the drain. Take it to your local recycling place. Don't get any on your face or lips (it's veeeery highly toxic).
2) Thermostat: If your rad is flushed and/or has nice new coolant in there (go for the pink stuff, since you're changing it) and the problem persists... it could be the thermostat. This is a simple bolt-on job (make sure you get the correct one for your engine.. they're very similar but different). CF1 is the more expensive of the two types.
3) Pump: If the flushed rad' and the new stat' doesn't fix it.. it could be the water pump. How old is your pump in age/miles? To change the pump you have to remove the cam-belt so it's worth changing the cambelt and tensioner at the same time.. which makes it an expensive job ... but at least that's that sorted for the next 36,000 miles.
Water pumps on the T/Spark can generally run to over 100,000 miles but if you change it every second cam-belt (i.e. 72,000 miles) then the beast will never go wrong. Leaving it for 3 cam-belts (108,000) might be a bit too much.