The peak can be climbed from the Uganda or Kenya sides, but as of late, the best route is from Uganda, even though it is a longer walk. I know only the Uganda side of the mountain, and thus this will be the side described.
To get to the trailhead, make your way by bus to Tororo and then Mbale. From Mbale catch a "matatu" (mini-bus) to Budadiri. This is the trailhead. There are hotels is Mbale and one in Budadiri. You must arrange everything for the climb (permits, porters, and a guide) in the national park service station in Budadiri.
A guide is usefull on the lower slopes, but not required. I found it worth the money to have a guide, or at least a porter who knows the way.
Day 1: From Budadiri (elevation 1100 meters-3609 feet), you will walk up an un-used and washed out road to Bumgabula villiage. The old road ends and the trail becomes very steep as you climb up through many terraced fields and along side houses. There are several trails here, so the guide here is really usefull. If you don't have a guide locals can point you in the right direction. The scenery is spectacular, and plan to take lots of photographs of the hillside villages. The people there were some of the most friendly I've visited in the entire world. Continue east on a steep trail up to the cliffs called "the Wall of Death", which is spectacular and exciting, but not as bad as the name implies. You will climb a few bamboo ladders here. The steepest part of the climb is over shortly after the cliffs, and from there on it is a very enjoyable trek through the jungle. Keep an eye out for Colobus and Blue monkeys; we saw several. Bear left at a junction. The Sasa River camp is at 2900 meters (9514 feet), and is a nice campsite.
Day 2: This is a short day to Mude Cave camp at 3400 meters (11,155 feet). The trail is easy to find, and the walk is pleasant. There are some nice views once you reach timber line.
There are several options for day 3-4, and even 5. You can spend all day making the trek to the Suam River Gorge, crater floor, and hot springs. This is 22 kms round trip (14 miles). You must be accompanied by a park ranger from the Mude Caves camp to visit the crater. There is a nice waterfall along the way. You can also make the trip to the highest point of Mt. Elgon which is 4321 meters (14,172 feet), and is named Wagagai Peak. If you're not up to it, you can climb the shorter Jackson Summit. You can also climb Lower Wagaigai at 4301 meters (14,111 feet). There are also some really beautiful waterfalls not too far from Mude Cave Camp, so if you have a guide, ask him to take you there. At the very least, plan on visiting the falls the same day that you reach Mude Cave Camp, the hot springs on day 3, and the summit on day 4. Another day can be added for Little Waigagai if you desire.
The route to the crater and hot spring will not be described in detail, since you are required to take a ranger with you. it's a very beautiful walk.
To get to the summit of Wagagai from Mude River Camp, follow the trail towards Jackson Summit, whick looks very similar to Sugarloaf near Rio de Jainero. The trail skirts around Jacksons Summit (you can make a detour and climb it if you like), and heads up the ridge to the south. After you reach the ridge, there is some scrambling to reach the summit. Enjoy the view if the weather is clear!
Days five or six will be spent trekking from Mude River Camp to Budadiri. This takes all day for most people.
The mountain is very wet, and can be very cold up high, so be prepared for this. There are no huts, as is the case with most other major peaks in Africa, so bring full camping gear. Rain gear and a waterproof tent are an absolute must, as are a good pair of boots.