You're bound to get several great responses detailing all these "must-have" recommendations but if you're just learning how to properly take care of a vehicle then it can all be overwhelming.
My suggestion, for what it's worth, is to ease into it. Find out if you really want to devote the time and effort needed to use a dozen products on a regular basis. This stuff can be expensive so you don't want to dump a bunch of money into it before testing the waters (like I do with most hobbies).
So maybe you could go down to the Walmart/AutoZone/etc and pick up a few things to get you through a couple washes, then you can decide what you might want to add to your routine.
Assuming you have an appropriate length of water hose, a nozzle, and bucket:
Car wash - The Meguair's NXT wash is pretty easy to work with and I think it leaves a nice finish. The big jug runs around $7-8
Mitt - I use the long-knit chenile microfiber something-or-other mitt (the bright green one) and like it. I've had it for 2 years so I'm not sure what it cost initially. I'd say probably also around $8
Drying - I've gone through several chamois over the years and don't like how they get slimey feeling after a few uses. The waffle weave towels I haven't tried yet because they look expensive. I've gone through 2 or 3 Absorbers, however, and they do a great job. I use the flat side for my top and upper surfaces then use the textured side for drying my cladding and bumpers. Just wring it out and stuff it back in the tube when you're done. Easy peasy. $5-6
Clay - The blue Clay Magic stuff really seems to make a difference, and it's a step you only have to do a few times a year. The lubricant also smells really good... :c) Basic kit runs about $12
Microfibers - Walmart/Target sells them in bundles of 4 for about $6.
Dressing - For tires I really like the Armor All gel in the orange squeeze bottle (comes with applicator). It leaves a sharp deep sheen without a greasy looking mess. It's also clean enough that it can be used on trim, whereas other tire products leave oily streaks. This will at least save you from having to get a seperate trim dressing for the time being. $4-5
Glass - The Stoner's Invisible Glass is an outstanding product. $4
Shop vac - This is an essential. Keeping the carpet (and seats) swept is a must to prolong the life and condition of it. I think you can get a small portable one for $30.
Polish - This depends on the condition of your paint. If you've got a newer car with good paint then I think you can skip this step for now. You might be able to get some minor swirling out doing it by hand, but unless your paint is oxidized & faded I don't think the effort required is really necessary. If you do need one, I think a good paint cleaner is probably your best bet so that you don't have to apply 2 or 3 different products over the entire car. Somewhere around $7-8
Wax - Hundreds of products all claiming to be superior to the others. For a starter product that's easy to find, I don't think you can beat the properties of Meg's NXT (I use the solid wax, not liquid). Use
very little because with a good foam pad this stuff goes on like butter. I think (and others have agreed) NXT leaves a beautiful finish. Take a look
HERE for a few shots. That's just a wash, clay and NXT with some armor all dressing, no polishing or anything. Can of NXT is around $8 I think.
Pads - I've found through my experience (or maybe I've just fooled myself into believing) that for polishes/compounds a microfiber pad is better at grabbing old paint. For waxing I think a foam pad does better at pushing the product onto the paint. A pack of 2 is $3-4
So like I said, that's one opinion out of hundreds here, but its at least a start. I just added it up and that's right at $100.
I bought a $200 PC on the suggestion of folks here but continue to do my cars by hand. I don't know why, I guess it's part of the bond. I'm not saying you won't want/ use one eventually, it just supports my statement that it's easy to jump in and get over your head with products and then find out you may not even use it/them.
Hope this helps some.