If you want to get to Kamloops from Edmonton, the fastest route is via Jasper and the Thompson River route. From Edmonton, you will head West to Jasper (haven't been this route since I was about 8 yrs. old, so don't know what the conditions are like now!). Once in Jasper, you can enjoy the beautiful scenery of Mt. Edith Clavell, the Marmott Range, and Pyramid Mountain! From there, you continue West to Tete Jaune Cache, and then turn South. The drive is gorgeous! Stop to take photos of Mt. Robson, the Pyramid Falls and Little Hell's Gate. You will then pass through Blue River and on to Clearwater. Clearwater is the gateway to Wells Grey Provincial Park, a huge park with lots of camping, hiking, fishing(?), and great photo ops! Continue south for another few hours, via Little Fort and Barriere, and you hit Kamloops! All in all, I'm under the impression that the drive takes from 8-10 hours, more depending on traffic and weather.
If you're travelling from Calgary, the fastest route to Kamloops (and to Vancouver, coincidentally) is via Rogers Pass in the Rocky Mountains. The trip is absolutely spectacular. Make sure you give yourself the whole day so that you have time to stop for photo ops!
The route goes something like this. Calgary to Banff is about 1-2 hours. There is lots to see in Banff, so take the opportunity to stop. By the time you reach Banff, you are in the heart of the Rocky range. From Banff, it's about 3 hours to Revelstoke, with Field and Golden on the way. Revelstoke is charming, take the opportunity to stop for lunch, and visit the lower dam just north of the town. From Revelstoke to Kamloops, it's another 3 hours. You'll pass by Craighellachie (spelling?), the site of the CPR's last spike, Salmon Arm, on the Shuswap lake, and Chase, on the Little Shuswap. Shuswap is a great place to stop, lots of campsites, and warm summer water.
Kamloops Transit is not very good. Because the city is so large for such a small population, the Transit Authority has really had to stretch it's resources in order to accomodate the most people. The problem is that the resources are small to begin with because nobody rides the bus due to poor service... You get the picture.
In addition, the routes have changed recently to include the local university, but time calculations weren't very well down, and now a number of the buses are often off schedule.
If you are not in any kind of hurry, the bus is the cheapest motorized transportation around Kamloops. If you are staying in the downtown core, the transit exchange is located at Thompson Park Mall. All the routes have stops here, so you can catch any one of them.
To all you non-Canadians, take heed: this tip's for you. The trip from Vancouver to Kamloops has evolved significantly over the years. It's most recent incarnation, via the Coquihalla Hwy, is a beautiful but very unhistorical 3 hour jaunt. For those of you looking to travel as the pioneers did in the 19th century, take the TransCanada Hwy from Hope to Kamloops. The trip is an hour longer, but there's no toll, and there are plenty of gems along the way. At Hope in the Fraser Valley, follow the signs for the #1 Hwy. Once past hope, you're in to the scenic Fraser Canyon. There are many great stops along the way for pictures and activities, including historic Yale, Hell's Gate tram, and the Ashcroft Teahouse. To get the full effect, plan to spend the whole day on the 4 hour route.