For lithium ion batteries, the nominal cell voltage is 3.6V. However, in practice the cell voltage will vary from 4.2V when fully charged to the usual end point voltage of 2.6V. They have no memory effect, and can be stored either charged or uncharged. However they lose storage capacity from the moment they are made - it used to be about 10% per annum. Many of them have mechanical, chemical and electronic protection built in, and the third terminal generally communicates in some way to the charger and/or the load, to regulate the current/voltage. However, you can make a simple constant voltage charger, set at about 4.1V per cell, and in many cases ignore the third terminal communications. At 4.1V , you will not get the full charge into the battery, of course. More than 4.2V will not work - either the protection built into the battery kicks in, or if none present, it explodes, or worse.
If you want a flat voltage discharge, and a simple charge circuit, Nicads are best, but they have memory, and don't allow them to fully discharge. The simplest charger for a more robust battery is for lead acid gell - Gate's Cells are pretty good, and monitoring the voltage on discharge will give an indication of the remaining charge. NmH cells are tricky to charge fully, and do not have the flat discharge curve (oxymoron?) of the Ni-cads.
The choice of battery technology depends on the type of load you will be using - Li-ion will shut down if you are powering, say, a circuit with large capacitors, or you will need some sort of 'soft start' since it sees the inrush curent as a short circuit. However, a cheap li-ion does not always have the inbuilt protection, so it will either be OK, or most likely it will self destruct, taking part of the local environment with it.