Both conglomerate and breccia are coarse-grained and poorly sorted clastic sedimentary rocks. However, the grains in conglomerate are rounded whereas the grains in breccia are angular.

These are both clastic rocks. Sandstone is medium-grained so you may be able to see individual sand grains. Shale is fine grained, so individual grains will be too small to see and it will feel smoother than the sandstone. Also, shale is distinctive in that it breaks apart in layers.

Travertine and limestone are both made of the mineral calcite. One way to distinguish between these rocks is to look for fossils of reef-building organisms in the biochemical limestone. Travertine is a chemical sedimentary rock, so it won't be built from the shells of marine organisms.

Both are chemical sedimentary rocks. Chert is made of silica (the same composition as quartz) and gypsum is a mineral as well as a rock. You could test the mineral properties of the samples (chert will scratch glass, but you can probably scratch gypsum with your fingernail), but you could also add water to the samples. Gypsum will dissolve but chert will not. (But don't do that if you want to keep your gypsum sample safe.)