The Rover K-series engine found in Series 1 and early Series 2 cars is pretty robust but over-revving may result in cracked cylinder liners. Check for an excessive amount of smoke from the exhausts. Also be sure to check when the belts were last changed. Toyota engines have cam chains, and are usually trouble-free.
Transmissions are tough, but the shift action may slacken with age. If so, the car may require a replacement cable. This isn’t an expensive job, but it is fiddly.
Elises are bought by keen drivers – some more talented than others. Be sure to check for accident damage, and for signs of excessive track day use (if you see gravel in places it shouldn’t be, chances are it’s been in the kitty litter).
There have been several recalls during the Elise’s lifetime so do be sure to check documentation to ensure that the car you’re interested in buying had the work done in period. If the car has little paperwork with it, you should probably look elsewhere anyway.
The use of composite materials and bonded/extruded aluminium works well, but corrosion to the footwell wasn’t uncommon in S1 and early S2 cars. Given the lack of carpeting, you will notice pretty quickly if this is the case.