Unfortunately, the train service between Kapiri Mposhi and Lusaka is pretty limited these days. It's only twice a week (departing Kapiri Mposhi at 16:42 on Mondays and Fridays, which doesn't connect well with the Tazara service), and economy-class only. I don't know the journey time to Lusaka. See my previous post at http://www.lonelyplanet.com/thorntree/thread.jspa?threadID=1962474 for more details on RSZ train services.
Although I would normally recommend train over bus, in this case bus is probably the better option. There are fairly regular express buses from Kapiri Mposhi to Lusaka, journey time ~3 hours, cost ~55,000 ZMK/person. Most of these buses start their journey further north in the Copperbelt, and just make a brief stop in Kapiri Mposhi as they are passing through. So, if you go to the bus station (on the main drag, near the 'old' railway station) you should be able to get on the next bus that comes through. I can't remember the names of the carriers, but there are several on this route. There are probably also minibuses doing this route (presumably slower and cheaper).
Note that the Tazara train arrives at the 'New Kapiri Mposhi' station, which is ~1km away from the bus station, down a side road. It should be max 12,000 ZMK for a taxi, or it's a longish walk. I suspect that when the Tazara train arrives, both the Tazara station and the bus station will be swamped with touts offering to help you find taxis, buses, tickets, etc.
Kapiri Mposhi isn't a huge place, but it has a bank (with ATM, I think Visa only), small supermarkets and cafes, internet cafes, hotel, market, and so forth. However, all this is along the main drag, there is very little out by the Tazara station.