1. I think the easiest way is to use a difference in solubility of lithium, sodium, and potassium carbonates (after addition of e.g. K2CO3, Li2CO3 will be the first compound in the precipitate). A synthesis of LiCl from Li2CO3 is relatively easy (just add HCl).
2. I know that it is possible to separate similar ions, like in the question, by electrolysis. The idea is that you have to force cations to move to the appropriate electrode and at the same moment you move the electrolyte in the opposite direction. The electrolyte movement has to be fitted to the ions velocity (mobility) - it should be exactly between it. Thus, a faster cation will go to the cathode whilst a slower one - to anode (because it will be taken away by the "faster" electrolyte). There is a similar experiment in students' laboratory on my faculty. This solution is of course more complicated but if you repeat such separations frequently, it is also worth of noting, I think.