Hello,

You will find the National parks website extremely useful for info on all the parks and campgrounds within them. You cannot just take your stuff into the woods and camp, you will have to camp in a designated area and/or get a permit for back country camping. If you are visiting a lot of parks you would be wise to purchase the annual pass for $80This entitles you to enter all NP's and Fed lands for one year but does not include camping and any tours etc, just entrance. The fee is equivalent to visiting 4 major parks, after that you are "in pocket".
I am sure someone will be along to advise you better on your travel plans than I, but most people, I am quite sure would prefer to start there journey in the North and head South trying to stay ahead of the extreme weather and below freezing night temps that you can see in Utah, never mind Alaska and Idaho !