Welcome Bill - is that one of your wooden ship models being used as your profile picture? If so, I'd love to see what the rest of it looks like!
Briefly, the laminating process is generally achieved by layering the printed copy (perhaps a former or some structural piece) to a thicker piece of stock (index, matte board, cereal box, etc) to add bulk/strength to the part. I personally like to use spray adhesive for this as it doesn't cause the parts to deform and I get even adhesion over the entire part. Also, I don't do my final trimming until after laminating. Check this thread out for great info on laminating:
Laminating template to cardboard
Purchasing thicker card stock: folks have to discussing a lot lately obtaining scrap matte board from framing shops for free (or nearly free). I collect scraps of tight grain, high quality cardboard as I come across them - this includes certain notepad backers, cardboard boxes and packing material (i.e.: tea boxes, food boxes, the cardboard that is sometimes stuffed inside a new shirt, etc). You can purchase cardboard in thicknesses of .5mm, 1mm, etc, but it's kind of pricey when there is so much that can simply be recycled. Just remember to laminate to the "clean" side...you don't want an otherwise perfect carrier flight deck ruined with a "Cheerios" logo ghosting through.
An inkjet is perfect - they vary a lot...some are genuinely better than others (ink chemestry, color trueness, etc). I'm quite happy with my $30 "disposable" HP-D2400 series as I find that the quality of paper is generally a bigger factor in image quality and color representation.
Best regards,
Adam