Separation of Binary mixture?
N.B.: the advice below mostly applies to organic compounds. If you're dealing with inorganic salts, or solutions thereof, the approach will be very different.

For a solid mixture, I would take an IR first. That should allow you to rule out potentially useless chemical tests by identifying the functional groups present in your mixture without wasting any of the sample. This would also give you some indication whether acid/base extraction is likely to be of any use. After that, it seems logical to test solubility in a wide range of solvents and at different temperatures. If you can find one that dissolves your mixture only partially at low temperature, but completely at high temperature, then you can likely separate the components immediately by recrystallization. Judicious use of a centrifuge can help if the crystals form a fine suspension.

In the case of a liquid mixture, I would again take an IR initially. In some cases, one component of the mixture can be chemically separated, and IR results would again be useful in determining the viability of that approach. Ketones and aldehydes, for example, will react to form semicarbazones that can be recrystallized and then identified by melting point. As far as physical methods of separation, standard are distillation and fractional freezing.