You can't just throw in a bigger breaker, and expect it to work the way you want.
First of all, you're going to have to have the utility install conductors sized appropriately for 200 ampere service. Depending on how the building is wired, you'll also have to upgrade the conductors that run from the service drop to the meter (this might be handled by the utility, depending on how the building is wired). Next, you're going to have to upgrade the conductors that run between the meter and the disconnect (likely handled by installing the new panel pictured). Finally, you may have to upgrade some of your grounding and bonding conductors, as they are typically sized based on the size of the service.
Once all that's done, then you can likely proceed as you've described. However, you're going to have to evaluate how the system is grounded. The service neutral will have to be bonded in the new panel, where the 200 amp disconnect lives. Then you'll have to feed the existing panel (now a secondary panel), with a 4 wire feeder (hot, hot, neutral, ground). You'll also have to isolate the grounded (neutral) bar in the existing panel, if it's not already.
Long story short, this is not a simple job. There are a lot of subtle details, that can easily be missed by a DIYer. I would recommend having an Electrician at least help you plan the project, and inspect the completed work.
To identify the wire gauge, the best way might be to take the cover off of the breaker box (advise to do this with power off for safety, but power can be left on - proceed at your own risk) and see if there are any indications on the wire sheathing. If there isn't, use a guide like to identify the gauge of wire.
As for copper vs. aluminum wire, aluminum might be cheaper in larger sizes. However, aluminum requires some special anti-oxident joint compound when you make connections with aluminum wire (see Ideal no-alox for an example of this stuff). If you don't use the compound, its possible that the connections will fail over time.
![No-Alox](https://i.stack.imgur.com/X392p.jpg)
EDIT: Using the info you provided in the comments, and the wire voltage drop calculator at http://www.southwire.com/support/voltage-drop-calculator.htm, assuming:
- Single Phase 240V
- Direct Burial
- no more than 3% voltage drop
- 2 AWG aluminum wire
Your wire gauge will allow you to maximum run somewhere in the 75-80 amp range. I wouldn't go all the way to 100 amps, but 60 amp service should work. You'll need 1 AWG wire to run 100 amps safely that distance.
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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.