That plate-under-the-gravel-and riser tube dealy is an Under Gravel Filter. (UGF)it is run with an air pump which is sold seperately.
I like this type of filter and it is inexpensive, But it is not so popular these days and you don't need it If you buy a Hang On Back,filter (HOB) the kind with a box for some filter media, and an intake tube that goes down into the water. This is the kind that would come with a kit, you see them all the time.
You do not Need an airpump if you are not running a UGF. If you like the look of bubbles, you can get an airpump, airline and an airstone, or some decoration for the bubles to come out of. But The aquarium doesnt need the bubbles.
I would say that you definitely need a gravel Vacuum. A net is handy,and a thermometer.
The only other expensive thing that I would say you Need is a Good test kit. The liquid type are much better than the strip type. If you absolutely cannot afford them, then the strips are better than nothing and you will want to have some of those. ammonia, nitrite nitrate and ph. Fish stores will test your water for you for free, and you can use that to double check your test strip readings if you have to.
Are you planning to put those rocks on the bottom instead of gravel? that would make it kind of hard to keep clean I think. You can use "wild" gravel if you wash the heck out of it. Rinse it till water runs clear. Instead of paying fishstore prices for gravel.
Oh, one more thing. Unless you use water from your own well, you will need something to dechlorinate the water. This is the ONLY additive you should need. Prime is Very popular around here. I used a less expensive brand that I can't remember. All it needs to do is remove chlorine and chloramines.
