Wiring Advice for Pole Barn
When digging a trench, conduit means you only dig it once, and makes future wire damage much less likely. Up to you, but often the price of conduit + wires is not worse, and sometimes better than, direct burial cable. I am also a big fan of putting in an extra conduit for future use, say if you decided to extend your network coverage to the barn. I prefer schedule 80 PVC in the gound unless I have a special consideration where rigid metallic conduit (RMC, it's like galvanized water pipe) makes more sense. In Barns, I prefer to keep using EMT conduit for the individual circuits, as rodents, wires, and barns can be a recipe for barn fires otherwise.

Aluminum wire is a perfectly reasonable choice for a heavy feeder - the modern alloys don't have the issues the old stuff used for 15/20 amp interior circuits did, and the cost is far less than equivalent copper, even though aluminum has to be a larger size. It is the standard material used for heavy feeders, though if you are unfamiliar with proper preparation for connecting it, it's worth having an electrician drop by to make those connections correctly (torque matters, anti-oxidant paste matters, and prepping the ends for the paste matters.)

For 150 feet at 100 amps with 75C connections and wire insulation, 1/0 aluminum is conservatively adequate. 90c would drop that to #1, but 90C wire insulation is much easier to come by than 90C rated connections, from what I recall when shopping. #4 is adequate for the equipment grounding conductor.

That information is not some magic I know, just me using one of the many web wire calculators that reference back to the NEC requirements. I would do the same for conduit size once a wire size/insulation type is chosen, though I highly recommend making the conduit bigger than the minimum size for your own ease of pulling - it's not very expensive to upsize conduit, and pulling near maximum fill is a pain.