Tough question, Khoob! You need to dissolve the powders in water, and then design a flowchart that will sort the ions in the different solutions based on their solubility characteristics. Here's a list of solubilities to test for:
Ions in Group I: Ag+, Hg22+, Pb2+-will precipitate in 1 M HCl
Ions in Group II: Bi3+, Cd2+, Cu2+, Hg2+, (Pb2+), Sb3+ and Sb5+, Sn2+ and Sn4+ -will precipitate in 0.1 M H2S solution at pH 0.5
Ions in Group III: Al3+, (Cd2+), Co2+, Cr3+, Fe2+ and Fe3+, Mn2+, Ni2+, Zn2+ - will precipitate in 0.1 M H2S solution at pH 9
Ions in Group IV: Ba2+, Ca2+, K+, Mg2+, Na+, NH4+
Ba2+, Ca2+, and Mg2+ will precipitate in 0.2 M (NH4)2CO3 solution at pH 10
Once you have narrowed down the possibilities for each powder, you could try a flame test to make identifications within the groups.
To test for the anions, there are some indicator based tests you could include at this link.