First the water issue is a bad one, water backing up to your house can float and damage the foundation. Ground water will also damage your patio. If you have a place to drain the water to, look into making a french drain. A french drain is a pipe with holes in it covered with cloth that keeps the dirt out, this in turn is buried in gravel. Then a normal pipe takes it to some place to drain the water off. I once had to make a dry well as I had no place to send the water. A dry well is a hole filled with rocks so the water can be diverted away from the house.
Now as to the cement work; if you have a pick up truck or can get one then you can get ready mix in the amount you need, it comes in a trailer. You can rent all the tools you need to do the work as well. You will need a large float on a pole and a small mag float, and an edger to round off the edges. You will need some 2x4's for the forms, and you will want a 12' 2x4 to smooth the concrete with. You will want some plastic to cover the concrete with, this keeps it from cracking in the heat. You may need a wheel barrel to move it to your patio. You will need some shovels to move the concrete around with.
I figure it will take about 1.75 yards of concrete to do it, so 2 yards should be a safe bet, better to have too much than not enough. You can plan to use the rest some place, I know people make stepping stones out of the extra. You could call and see if some one will bring that small of an amount; 2 yards, but the also charge for the time they sit as you work. You could do it with bags but that is a lot of bags and a lot of mixing, rent a big mixer if you do.
The concrete should be 3 and 1/2 inches thick, you can go as high as 4", 3 1/3" is the hight of 2x4's so this works out good. Lay the 2x4's out in the shape you want. Make sure they are level, square them off by measuring side to side, then measure diagonally like in an X pattern, corners to corners, when both measurements are the same you are square.
Pound stakes into the ground on the out side of the 2x4,s to hold them in place and nail a board across the corners, not over the cement area.You can use 8' ones, just nail a cut off chunk of 2x4 to the out side of where the meet.
I recommend putting in the wire mesh to keep the patio from breaking up, you can get it at any big box hard ware store in the size you need. They sell little blocks to set it on.
You will want some help if you can get it. I start from the back and work to the front. When the concrete is poured take a the 12' 2x4 and with the help of some one smooth the concrete until it is level. Then take the float with the long pole and work the concrete until it looks good, smooth the corners with the small float, and the go around the edges with the edger. Let sit and until it sets up, cover with plastic. If you want the brushed look. when it is set up in about one hour or less take a push broom and go over it. Use a strong plastic kind as you do not want the bristles to come off in the concrete. In a day or two you can remove the forms, stay off of it for a week if you can.
I typed this with a keyboard and my iPad, next time I write so much it will be on a real computer, it is hard to see and get things right on the iPad.
The things I do are an evolution and I am always learning. My way is not the only way of doing things, and I may and will change the way I do things as I learn better ways. So any advice that I give is in that spirit.