I map this trip out every year for fun.
The 2015 schedule is approx. 31,600 miles of traveling - probably $15-17,000 for fuel and routine maintenance such as oil changes. May require new tires during the trip is you don't start with new ones.
Some races are going to be very difficult to reach on time - some near coast to coast drives in just seven days.
You will have to contact individual tracks for tickets. Some brokers might have a way to create a full-season package for you. 36 races plus the All-Star and the Daytona festivities are going to average $50-120 if you only go to the race Sunday, $120-400 if you go to all the weekend races - say $3,600 per person for tickets for 38 events.
Campsites can be expensive because many campgrounds will require you to book for the entire week of the race. They won't take anything less. Some tracks such as Martinsville dry camping are relatively cheap - about $100-150, some will cost $300-500 for a week. Hit the middle $250 per race weekend - $9,250.
The 2015 schedule (Date of race / track-location / miles needed to get to the track)
2/22 - Daytona 500
3/1 - Atlanta, GA - 415 miles
3/8 - Las Vegas, NV - 2,000 miles
3/15 - Phoenix, AZ - 310 miles
3/22 - Fontana, CA - 320 miles
3/29 - Martinsville, VA - 2,415 miles
Off Weekend
4/11 - Fort Worth, TX - 1,152 miles
4/19 - Bristol, TN - 990 miles
4/25 - Richmond, VA - 330 miles
5/3 - Talladega, AL - 640 miles
5/9 - Kansas City, KS - 750 miles
5/14-24 - Charlotte, NC - 1,000 miles
5/31 - Dover, DE - 510 miles
6/7 - Pocono, PA - 170 miles
6/14 - Michigan, Jackson, MI - 545 miles
Off week
6/28 - Sonoma, CA - 2,315 miles
7/5 - Daytona, FL - 2,935 miles
7/11 - Kentucky, Sparta, KY - 890 miles
7/19 - New Hampshire, Louden, NH - 915 miles
7/26 - Indianapolis, IN - 950 miles
8/2 - Pocono, PA - 670 miles
8/9 - Watkins Glen, NY - 180 miles
8/16 - Michigan, Jackson, MI - 500 miles
8/22 - Bristol, TN - 585 miles
Off Week
9/6 - Darlington, SC - 325 miles
9/12 - Richmond, VA - 305 miles
9/20 - Chicagoland, Joliet, IL - 810 miles
9/27 - New Hampshire, Louden, NH - 995 miles
10/4 - Dover, DE - 445 miles
10/10 - Charlotte, NC - 510 miles
10/18 - Kansas City, KS - 995 miles
10/25 - Talladega, AL - 750 miles
11/1 - Martinsville, VA - 465 miles
1/8 - Fort Worth, TX - 1,155 miles
11/15 - Phoenix, AZ - 1,070 miles
11/22 - Homestead-Miami, FL - 2,415 miles
I've thought it might be possible to do in two or three years by breaking the country into regions, or possibly in a year by skipping some second races at certain tracks.
The Sonoma to Daytona and the Phoenix next to last race and the final race at Homestead is real long drives. You will have to do 400-500 miles per day for five or six days straight. You probably won't be able to do more than a 8 hour stop at Walmart on travel days. That will save some money.
A couple years ago, I figured that it would take $45-47,000 to be sure of completing the trip - and fully half the money would have to be spent in advance with no refunds if something happened and we were unable to make it through the whole circuit.