I have a 400 amp service in my home and will be running a line to my shop with a total run of 250' to feed a sub panel 125 amp service. What wire size is the 250' run? Would a 100 amp service be the same wire size?
I would like to run copper wire, is aluminum that much less expensive? Any other reason to use aluminum over copper?

Use #4/0 Aluminum XHHW wire for this run. I would use 3 " PVC Schedule 40 conduit and pull 3 #4/0 XHHW for the 2 hots and one neutral, and 1 #2 XHHW for the equipment ground.

You can price copper, using 3#2/0 XHHW or THWN and #4 XHHW or THWN.

Along with the high cost of copper for this run, be prepared to pull the weight of the copper wire.

Only issue with aluminum wire is to be sure to treat the connections properly, see the attached recommendations by a manufacturer.

You can use direct buried URD cable, but depending on soil conditions, a conduit will offer more protection, and allow a defective cable to be pulled later on.

I checked on the #4/0 aluminum wire and the diameter is more than the breaker accommodates. Is there a work around for this?
A 125amp Siemens breaker to go into my homes service panel.

You can reduce the wire size by splicing to a smaller #2 or a #1 at the both ends.

you can reduce the wire size by splicing to a smaller #2 or a #1 at the both ends.
Things are coming together. I found an adapter from Ilsco that reduces the 4/0 aluminum to a 2/0 copper so it will fit my breaker. (Ilsco part CPM-4/0) ($15. Each)
I bought three just in case I need one for the neutral also. (I have not looked at my neutral bus bar yet to see if it will accept another 4/0 aluminum.) If it will not, is it OK to use one of these adapters on the neutral? These adapters are to be used on the feed service end of the line.

The Ilsco CPM crimp is perfect, and will be needed for the neutral. The special crimping tool and die will be needed to crimp these connectors onto the wire.