Hello all

I've got a shallow well (35ft) at my home with a submersible pump and small pressure tank. The pump burned up last spring, and at the time my landlord had someone come look at it (i was renting the home at the time). Doing that, he pulled the pump and said it was bad, and threw it away. My landlord never did anything to repair the well, so it became my problem after I purchased the home from him.
I'm just now working on replacing the pump, and I'd like to consult with some others who have worked with these systems more than I have.
Basically, i'd like to make sure the pressure tank is connected to the well properly so the pump doesn't cycle as much as it did before it died. Also, I should note that this well is just for outdoor uses only, it does not feed the water supply in my home.
Anyway, when the well was working, the pressure would rise and drop, I could watch my sprinklers slow to a trickle and then pick back up when the well kicked on and built pressure. After a minute or so, they'd slow down again and the process would repeat itself.
Upon removal of the piping in the well, I noticed that there was a check valve between the inlet to the pressure tank, and the outlet to the hose connection. I removed the check valve to see if it was still good, and sure enough, it was stuck open. Is this check valve necessary? I know the new pump has a check valve in it, is this enough to keep the pressure in the tank? The pressure tank was replaced a couple of years ago, and it looks to be in good shape. But I have not checked the bladder to see if it will hold air. In fact, the pipes have been cut since the well was removed last spring.

I'm hunting for information because I'd like to do this right once, and not over and over again. I've already got new wiring installed, and I am to the point that I'm ready to connect the pump to the well pipe and drop it in. I just want to make sure everything is good before I finish.
If you need more clarification, or pictures even, let me know. I've got all weekend (fri-sun) to work on this.