I use this material quite often for my work (building theater scenery). My advice is just echoing what others have said here.
Partially it will depend on what you're cutting it for. Buildings with a straight edge? The table saw is the best way to go, but please be careful. A bandsaw works wonderfully for cutting curvy circles and hill shapes. A jigsaw (saber saw) should work pretty well as long as you've got a nice sharp blade.
A router also works well for cutting and carving, as long as you've got a decent jig. I've made 12" wide 10' diameter circles out of laminated overlapping sheets, and a router on a stick was the only tool used.
I'm also quite lucky in that my workplace has one of these: Original Foam Cutter by Riema
It is Deluxe
. I cant imagine you'd shell out $200 for one, let alone $500 for the fully kitted out set, but it's a fantastic tool.
I'm a much bigger fan of this knife style foam cutter than the bow style cutters, but they both have different applications.
I've used this knife for making hills and caverns for my minis and just love it.
If you don't have a wood shop, however, let alone a scenery shop, a hobby knife should work well. The trick is to not press too hard. take multiple passes. Trying to cut all the way through will rip the foam and give you crappy edges.
The material is typically intended for insulation in houses, so it's rare to find it in less than 2'x8' sheets, or even 4'x8'
A hardware shop might be able to cut it into more manageable sizes though with their tools, e.g. 2'x2', if you ask them.
It does however sand quite nicely and smoothly. Sandpaper will get you there, although a light touch with an electric palm sander or maybe a hobby dremel would speed things up. We've made full sets of snowscapes before out of the stuff, with lots of sloping curves.
Be careful of the particulate dust, try and work outside. Your wife will thank you. And if you use any kind of hot tool, definitely work outside or in a well ventilated space. The fumes are not at all good for you and you will feel it.
A couple quick words about gluing? (as though i haven't ranted long enough) The ideal for foam is a two part contact adhesive. I don't remember brand names off hand, but the stuff is awesome. We use it at work for applying foam to walls, etc. However, for light use.... wood glue or white glue will actually work. It takes a long time to dry, because there is very little airflow in between the materials.
A speedier and smellier alternative, is to use a spray adhesive. In the states the kind i've used is called spray 77. (spray 90 is a heavier duty version). If you use a very light coat on both faces, it won't eat into the foam and you can get a really good fast stick. if you spray too heavy, the thing will just disintegrate in a pile of nasty fumes.
As mentioned the same is true of spray primers. The acetone in spray cans will dissolve pink , blue and bead foam. Acrylic or even latex (house paint) is your friend.
If you've got a ton to do, you can find a pump sprayer (hudson sprayer) use thinned and filtered latex paint . Brushing it on might be just as fast....
That's my input... hope its not too unclear....