Of the various ways to discover and characterize exoplanets, two photometric methods are feasible for typical well-equipped amateurs: measuring the intensity dip as an exoplanet passes in front of its host star and measuring the brightness increase as it and its host pass in front of a background star. The latter technique is, in principle, capable of detecting exoplanets in external galaxies!
The British Astronomical Association has recently set up a sub-section to encourage and assist observers of exoplanets and much useful material will be found at their Exoplanets Division page and links therein.
When a planet transits in front of its star it blocks a tiny fraction of the starlight. The fraction is rarely more than 3% and often under 1%, corresponding to a magnitude drop from 0.03 to less than 0.01 magnitudes. Variable star observers are familiar with how to measure the brightness of stars, so little more will be said here about equipment and data analysis, other than to note that a camera is essential and that very careful attention must be paid to data processing.
A typical transit lasts for a few hours and observations should begin well in advance of the predicted start of the transit and continue to well afterwards. An important use of these observations is to pin down the time and duration of the eclispes, to within a minute or so, in order that the ephemeris can be made more accurate. Predictions are sometimes 30 minutes or more out from reality. At the time of writing (2021-09-10) I have observed two transits: HAT-P-49b and WASP-65b.
When a massive object passes in front of a star (or galaxy for that matter) the former's gravity deflects and magnifies the light coming from the latter. Even a planet the size of, say, Neptune, can increase the brightness of the star by several tenths of a magnitude. This is much easier to measure than a 0.01 magnitude transit! A rather nice example of amateur observations of gravitational lensing arising from an exoplanet may be found here.
Until recently it was difficult to find reports of an otherwise unexplained rise in brightness of a nondescript star. With high-cadence all-sky surveys coming on-line, alerts are now issued relatively often — notably from the Gaia mission. The Gaia Science Alert Project at Cambridge University looks for such events and makes them available. Roger Dymock, the BAA Exoplanets co-ordinator, monitors these events and mails out a list of gravitational microlensing candidates weekly. Much more information is available at the link given above.
A benefit of observing gravitational lensing candidates, compared with transits, is that it is rare for any one to be observed more than daily. It is then possible to fit in several into ones regular observing program.
A significant difference in practice between observing exoplanetary events and classical variable stars is that the latter generally have readily available and high quality lists ("sequences" in the VS jargon) of nearby comparison stars, their positions, their magnitudes and an estimate of the measurement uncertainty of the latter. This is rarely the case for host stars of exoplanets, especially not for those not yet known to host any exoplanets. I have tried to alleviate this difficulty by writing some scripts (one of which may be found on the software page) to create sequence files and to make the results generally available on the exoplanetary sequences page.