Any suggestions on where to begin (as a beginner) on a boundary water canoe trip?

Is there a two day - no camping alternative?

Where?

You should begin by Googling "Boundary Waters canoe trip" and reading, reading, reading. All the information you could possibly need to start the process will appear.

If you don't want to camp you will have to stay at a resort outside of the park. There are quite a few near Ely.
See
http://www.northshoreinfo.com/lodging/ely.htm

The more remote areas take several days canoeing to get to. An alternative would be to go to a resort in Canada on the border. Some Canadian border resorts you fly into. From these you could access some of the more remote areas. In my opinion Lac La Croix is one of the best lakes. There is a waterfall near the eastern part of the lake which is fairly impressive.

You can drive right to Gunflint Lodge and stay in lodge/cabins right in the BWCA. It is a bit spendy. They have a restaurant on site. We have used Hungry Jack Outfitters camper cabins (basically a "plywood box" with bunks) the night before we go out on a trip and it is spotlessly clean. No meals available there, however. They also have 2 full service cabins for rent...but I think in most cases you will need to rent a cabin for more than 2 nights. You can drive to MANY campgrounds in the area...we have stayed in a tent at Flour Lake with our car.
You have oh so many options. Do a little research online and call a few lodges/outfitters to see what interests you. My DH likes to go out of Ely, MN due to his nostalgia, I prefer the Gunflint Trail because I like to go through Grand Marais and see Lake Superior. Happy Trails! It is one of the best places on the planet, IMHO!

I totally agree with Aunt Annie, I too love to go to the Gunflint Trail. My very favorite resort is Gunflint Lodge and as Aunt Annie says it is expensive buut there are tons of other outfitters and lodges on the Gunflint Trail and I am sure one would be perfect for you. We also love Grand Marais and that is one reason I much prefer The Gunflint Trail to Ely but lots of people love Ely too. Look under the Gunflint Trail Association and you should get some help there. Also, if you do go there, don't miss blueberry pancakes at Trail Service Center some morning. YUM!!

Stay at a resort or motel in Ely. Get a day-trip permit for entry point 30, Lake One. This lake is fairly large with lots of bays and islands to explore. There are no motors allowed on this route so you don't have to worry about boats. If you wish to explore further and have the time, you can portage into Lake Two, Three, and Four (all connected). To make this easier, book your stay at Kawishiwi Lodge which is right on the end of Lake One.