Copper is very ritzy and
very expensive. If price is not an issue, and you're going for a look
like this, then you can consider copper.
I've always used aluminum. Couple years ago I checked into vinyl gutters for a job and had a hard time finding someone to install them. Most said they're junk.
As for continuous vs. sectons:
With continuous, you have just one piece of gutter for the whole roofline whether it's 20' long or 80'. They put it up in a jiffy and it's easy to get the slope right.
With sections, diy'ers buy a bunch of lengths, attach them one at a time, work hard to seal the seams right and find a hard time getting the slope right for each piece.
With continuous, the job is done in a 10th of the time.
You'll find getting estimates is easy. The price will simply come down to how much per foot...and chances are, most gutter contractors in your area are using gutters from just one or two manufacturers.
), it is difficult to find a company who will even pick up the phone. I actually had two companies tell me they aren't accepting any new work until they catch up to their current workload, with a five month time frame given. 
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