Welcome to the forums. The Schwinn Breeze certainly qualifies as 'Classic & Vintage' –you'll get more & better help if you post inquiries there.
The Schwinn Breeze of the '60s and '70s has a Schwinn S-6 rim. The bead (tire flange) diameter is 597 mm. Available at Schwinn shops and other local bike shops, and many online sources for these tires including at Harris ( http://www.sheldonbrown.com/harris/tires/597.html
). Kenda is the most common brand, I think there's only one other. Two Kenda styles – gum & blackwall. These originally came with blackwall tires. Look for '597' and/or 'S-6' and/or 'EA-3' in addition to 26 x 1-3/8.
Don't get a 26 x 1-3/8, '590' tire. Almost all other makers' bikes use this size, but not Schwinn.
( !!! If this is a Schwinn Breeze of the 1980's -- discount the above advice about tire size! Look for markings on the tire and/or rim -- may be a 590 then. After Schwinn closed the Chicago factory, much about the bikes changed. )
How to tell if the wheel (note: wheel= rim+spokes+hub; wheel does not include the tire) is damaged beyond repair? If it wobbles side to side or up and down in a very gradual way, and only 1/4” or so max, as you slowly spin it, then my guess is it's fixable by 'truing', adjusting the spokes. YMMV. If the wheel has short-period, pronounced hops or wobbles and big dents (i.e., the metal is badly bent) as if it was run hard into curbs or potholes, it may not be practically fixable. And, if the wheel is badly damaged, check the frame, fork and headset for damage and misalignment too. (That is, check to see if it pulls to one side and/or the steering feels 'funny'.)
If you want to buy new wheels, I am not aware of any available in the 597 size. You can probably find used Schwinn steel wheels if you're patient; these bikes were made in the millions and with the solid Schwinn quality a great many are still around.
For an upgrade, go with aluminum alloy wheels. Again, 597 wheels are not available new, but 590 wheels are usable (and then you'll need 590 tires). The brake shoes will have to be lowered 3.5 mm. You may or may not have enough travel in the caliper slots to do this. This would be costly though – at least $100 I'd guess and you may need to have the rim rebuilt around the present rear hub -- plus more time & money to find replacment brake calipers if you don't have an extra 3.5 mm. (Never mind the advice about caliper brakes if the bike is equipped with only a coaster brake.)
It's around 30 years old or older. These Schwinns are good, solid bikes, but after all these years the minimum service needed is to replace the old-technology grease which has long since hardened, and to true the wheels which probably have dangerously loose spokes. These Schwinns are easy to work on, needing relatively few special tools to completely service. I'd encourage you to go ahead & learn to do it yourself--or employ a handy friend. Lots of resources here and elsewhere on the Web. Have you found our late friend 'forum-don' mechanic extraordinaire Sheldon Brown's site ( http://www.sheldonbrown.com
)? Lots of books available—there was recently a thread here on that. Popular brand of bicycle tools ( http://parktool.com
), they have advice for home mechanics too. And, Youtube has bicycle mechanics demonstrations. Not that hard!
Good luck, please come back & let us know how it's going!