You are planning almost EXACTLY the same trip as another member (
click for thread).
My previous advice for LV to Yosemite is still valid.
1. There really aren't any "cheap" places to stay near major national parks. You would probably have to get a campsite to legally and safely sleep in your car, so you might as well bring basic camping gear.
5. Yosemite to Bryce can be driven in one very long day. It's 530 miles and allow about 12 hours. Take CA-120 back over Tioga Pass and stay on that to US-6. Fill your gas tank in Tonopah, last chance for about 200 miles (if you can't make 200 miles, it's 160 miles to Alamo, which has gas, but a bit out of your way). At Warm Springs, take NV-375 (the ET Highway). Stop at the Lil Ale' Inn in Rachel for some Area 51 kitsch (NO gas available but there are a few rooms in trailers for rent and they have food). When you reach US-93, turn left and go through Caliente. Gas and a few motels available. Turn right on NV-319 which becomes UT-56 to Cedar City, gas and major hotels available. Then take UT-14 to UT-148, go through Cedar Breaks, turn right on UT-173 and take that to Panguitch, gas and motels available. 20 minutes to Bryce. Maybe a bit much for just one day, but you have some enroute options for motels/hotels.
From Bryce, take US-89 south to UT-9 to go into Zion via the east entrance. About an hour and a half.
North or south rim - the north rim is a lot more remote and less crowded. The south rim is where everyone goes.
Zion to north or south rim - 2 choices. Leave Zion via the east entrance, go back to US-89, take that to Kanab, then Alt-89 to Fredonia - OR - take UT-9 west out of Zion to Hurricane, then UT-59/AZ-389 to Fredonia. Alt-89 to AZ-67 for the north rim, or stay on Alt-89 to US-89 to AZ-64 for the south rim.