Each state might be a little different but in Wisconsin, you start with the state Department of Transportation, Bureau of Aeronautics. They sent me a package that included all of the state and federal forms along with instructions. Both the state and federal forms were returned to the state. The state reviewed the federal forms for errors and then forwarded the federal forms to the regional airport district office of the FAA. The FAA contacts any airports near your location to see if they have any objections due to traffic pattern interference. These approvals were pretty straight-forward. It took about two months to get the state and federal approvals and there were no fees involved. (The FAA is mainly concerned with possible interference with other air traffic. The state was mainly concerned with interference with other forms of transportation such as highways and railroads that might be near your approaches.)
I received the state and federal approvals before ever talking to any local officials. (I later found out that the state had notified the local town that an application for approval of a private airstrip had been received.)
After the state and federal approvals had been issued, the real fun began: dealing with the local town planning commission, town board and county planning department. They required a formal plan of operation and approval of a conditional use permit. There were a lot of meetings and fees but eventually I got the required approvals. One thing that helped me was that I went around and talked to all of my neighbors before they got their certified letters from the county inviting them to a public hearing. I gave them copies of my plan of operation and assured them that they weren't going to have an international airport next door. All were very supportive and no one said anything negative at the public hearings. The town officials could hardly believe the lack of objection, but I explained that I had already talked to the neighbors which really "greased the skids".
The only person to actually visit my property was the county land use planner. He was very supportive of my plan.
I had the most problems with the town attorney who wanted to put too many restrictions on the airstrip such as no ultralights and no helicopters. I eventually got those two restrictions removed but it involved another round of public meetings.
Going through this exercise was a hassle but now that it's done I feel that it was worth it. The entire process took about 6 months. My grass strip is 1,700 feet long with pretty good approaches.