1. The "feel" of the specimen (the touch test).
2. Response to heating

Touch is helpful to categorize the thermal conductivity of a specimen. You can tell a lot with an experienced touch, or (especially with a small specimen with a glossy surface) you can also breathe on a cool specimen in your warm hand. If it is insulating, your breath might condense on the surface. If it is conductive, the warmth of your fingers may rapidly evaporate the moisture.

Insulating

These minerals are "warm" to the touch, and conduct heat poorly compared to most minerals. Well known examples are gypsum (which is noticeably warmer to the touch than a similar specimen of quartz), also sulfur, barite, and apatite. There can be some oddities. For example, biotite and muscovite have average conductivity along the sheets of the crystals, but are insulating normal to those sheets. The difference is much more dramatic in graphite, approaching diamond along the plane of the crystal, highly insulating normal to it. In reality, many minerals have different thermal conduction on differing crystal axes, but generally not so great as the micas let alone graphite.

Average / normal

Unfortunately, most minerals have a thermal conductivity that can't easily be distinguished by hand. These include common minerals such as quartz and calcite.

Conducting

These minerals are "cold" to the touch, as they rapidly conduct away the warmth of your fingers. This includes most metals, but note that diamond has the highest thermal conductivity known (five times higher than $#2, silver). There are a few other non-metallic minerals with anomalously high thermal conductivity, including corundum (sapphire & ruby), hematite, spinel, and pyrite.

Response to Heating: Some minerals have characteristic responses to heating, including:

Decomposing

Cinnabar easily reduces to metallic mercury, and galena reduces to lead (as the sulfur burns out of the compounds). Other minerals decompose by losing water of hydration (gypsum turns to anhydrite, borax turns into tincalconite which in turn dehydrates to powdery anhydrous borax, not a recognized mineral). Likewise, tremolite dehydrates into diopside.

Melting

Minerals with a low melting point include mercury, sulfur, zinc, tin, & lead (all elements).

Curling or Peeling

Some of the phyllosilicates (a subclass including clays and micas) are known for leafing when heated. The mineral Pyrophyllite (fire-leaf) is named for its ability to exfoliate into a flaky mass when heated. Likewise, the name apophyllite means "to leaf apart". While not a phyllosilicate, the mineral colemanite also exfoliates when heated.

Expanding / Swelling

Vermiculite is known for extreme expansion when heated (the vermiculite used in potting soil mixtures has already been expanded).

Other chemical changes

Amethyst can be turned into citrine by heating; this involves a change in oxidation state of the iron impurities which give both their colors.

Other characteristics:

Minerals containing sulfur often smell of sulfur dioxide when heated; minerals containing arsenic often smell of garlic when heated.