power to garage

Personally, I would run #2 aluminum USE (underground service equipment). It's relatively cheap and can handle 100 amps, although it's only good for about 50 amps over the distance you're looking at. Run 4 conductors: 2 hot, 1 neutral and 1 ground from your service disconnect to a panel in the garage. Ground rods at the garage; neutral and ground are isolated in the garage panel. End result is you will have 240 volt service in your garage good for 50 amps, enough to power a welder if everything else is turned off.

This project is going to be a lot of work. Do right and you'll only have to do it once.

I'm not aware of any code provision that would prevent you from joining different size wires as long as some mechanical problem doesn't prevent it. (Which is not to say that there might not be some provision against it.) But connecting the #8 to the #4 is putting your conductors in the wrong order anyway. Start with your largest capacity conductor nearest the electrical source. Smaller gauge wire upstream from larger limits the carrying capacity of the whole run to what the capacity of the total length would be if it were all the smaller gauge wire.

Calculating the voltage drop for a 180' run using three sizes of wire will be more complicated, but I'd advise you to do it. Type "voltage drop" on Google and you'll get plenty of calculators. I've never checked to see if they handle this situation.

All the best, Stan

The current doesn't care whether you put the #4 wire ahead of the #8 or behind it. The same current is traveling through the whole length of the conductor. You'll get X volts drop across the #4 section and Y volts across the #8. Total voltage drop = X + Y. (or Y + X, for that matter).