There are several steel trailer manufacturers that can supply you with a 2H trailer. If you purchase a used trailer, you can find many with upscale furnishings at similar prices.

My barn owners told me to stay away from aluminum trailers because they're hot in summer and they're even hot in the winter.

Any metal trailer will absorb its surrounding temperatures, AL or FE. Insulation will slow the transfer of cold, but ultimately the trailers' temperatures will reflect the outside temperature. Aluminum transfers the heat or cold faster than steel, but over time will end with similar results.

One difference is a bare aluminum roof. Bare aluminum has about the same reflectivity as black, and will absorb a great deal of heat, more so than painted steel. By coating the aluminum with paint or an Elastomeric product, this problem is eliminated. There have been complaints by manufactures of trailers that supply wooden flooring, that aluminum absorbs more heat from the road's surface and the wood acts as an insulator. This may have some validity as wood has an "R" factor for every inch of its thickness. On the other hand, steel will also be similarly affected by the road's heat, again offering little difference between the two materials.

I believe your barn owner erred when he said the an AL trailer would be hotter, especially in reference to winter hauling. It doesn't seem logical, and in my experiences, not realised. Aluminum trailers have many advantages over steel, which is especially relevant depending upon where it will be used. Steel is a proven material and has served many people for a long time. You have to decide what your budget can purchase, and pick the best trailer you can find within that constraint.

Any opinions on the brands Calico, Delta and or CM? 

At this point I don't blame on doing long hauling with my horses - at this point I just want something that will get me to my boarding barn to lessons back to the boarding barn, and maybe to Tulsa and back which is like a hour away (pulling a trailer).  At this time I don't show - so I am thinking just a starter trailer would be good if you will.