This is such a great idea!
As far as a budget goes I might be able to help. (Caveat- obviously this depends on your class size)
1.) Jars, Lids, and rings: Right now you can get a 12 pack of jam jars for $7-8. I’d recommend checking out Meijer (Midwest), Walmart, Aldis, or Costco.
2.) Canner- since you’re doing cranberries (which I’m sure you know as a science teacher have a low pH) you don’t need a pressure canner. So any tall pot will do! I personally use a large metal stock pot- You might be able to borrow some from your cafeteria kitchen, your own storage, or pick one up from Salvation Army for a few dollars.
3.) I’d really recommend getting a Jar Lifter. Especially if kids are going to be around boiling water- much less stressful than using tongs.
4.) Rack Replacement DIY- you can improvise with putting a washcloth/hand towel in the bottom of the pot. My mother and grandmother have always done it this way- you just need something that will prevent the jars from rattling (against each other or the bottom of the pot). A soft washcloth spread out on the bottom has always worked very well.
5.) Misc. Items- a few hand towels to help when you tighten the rings, and paper towels for wiping off the rims of jars
An Idea: (I’m not a teacher or anything) but it might be cool to use a jar or two to show your students some improper canning techniques. You could turn a jar upside down and explain that this creates a false seal and isn’t safe, you could fill a jar completely (leaving no headspace) and let it burst/leak through the lid.
This ended up being far longer than I intended lol
The site katanas35 links to has all the basics. But to answer your question, you can do tomato sauce and salsa in your pot, but for stock you would need a pressure canner. Pressure canners can be hard to find right now, and not really cheap, you might consider trying boiling water canning for a while, then if you find you like it and want to keep doing it, invest in a pressure canner.
Make sure you learn what to do and how to do it, and follow a tested recipe (all of which can be done at that site) to make sure your food is safe.
As for your pot, as long as it is tall enough to cover your jars with a couple of inches of water, it is big enough. I'd imagine your 5 gallon pot is plenty tall. You will also need
If I were you I'd actually just go straight for an inexpensive pressure canner and skip the pot altogether, the pressure canner can be used as a water bath also (Don't tighten the lid down like you do for pressure canning, just sit it on top or sit another lid on top) so it's all in one type of thing, less bulky appliances and less money spent along with a larger variety of things that can be canned.
I also recommend grabbing a jar lifter, this is an inexpensive one, doesn't have to be pricey and a funnel comes in handy for filling the jars, but you might already have one that works, if not you can get by without, make sure you wipe down the rims of your jars (before putting the lids on) no matter if you're using a funnel or not (don't want any unnecessary failed seals).
You don't necessarily need a book, you can use websites like The National Center for Home Food Preservation, recipes from Bernardin, and Healthy Canning, but if you do want a book the Ball Blue Book is a good place to start.
I know there's so much more info I could offer up, but I'm trying not to overwhelm you and hope this helps, good luck and have fun!
Look up canning jar handle
Norpro Canning Jar Lifter https://www.amazon.com/dp/B000HJBFGC/ref=cm_sw_r_apan_glt_fabc_2GTGZYCK55ACHRPNAJZH?_encoding=UTF8&psc=1You
In my experience pickles, chutneys and jams (with added pectin) are the easiest and have the best taste-to-effort ratio. These are all also high in acid by their nature, so you can feel very secure that they are safe.
Think about:
Something that's really fun about canning is that you can make things that you can't buy or would be very expensive. Examples would be: pickled green beans or asparagus, peach chutney, plum jam (from your aunt's tree), Grandpa's spiced pear jam.
Also, buy a jar lifter. I found mine used at a thrift store. I hate buying specialized gadgets but a jar lifter makes things so much easier and safer. Also be on the lookout for a really big pot and something like a cake rack that fits into the pot to elevate your jars so they aren't sitting directly on the bottom. Your jar size will be limited to whatever size fits in your pot sitting on the rack with 2in space left at the top.
Also, buy a jar lifter. I found my used at a thrift store. I hate buying specialized gadgets but a jar lifter makes things so much easier and safer.
I'm a little late, but something like this would probably make your life a lot easier - https://www.amazon.com/Norpro-600-Jar-Lifter/dp/B000HJBFGC