Since I was young I always wanted to go camping. I wanted to

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Since I was young I always wanted to go camping. I wanted to do like straight up in a forest clearing or lake shore or something of the sort, no campsite hook ups or anything of the sort. In other words I wanted back country camping.

I'm currently building an 8ft camper trailer which I plan on taking on cross country trips, hitting national parks and historical landmarks all along the way. Being that I live on the east coast I plan on doing many important US history spots, things like the Liberty Bell, White House, etc.

I've read that you can camp almost anywhere on national park or BLM land as long as you're 200+ feet away from any water source. How difficult would it be to find viable and hopefully scenic back country camping spots? Will there be separate exits on the highway for people trying to just get into the thick of the wilds of the park or will I have to find a trail or something of some sort that'll take me from the developed park area to the back country areas?

I'm definitely trying to avoid getting lost in the wilderness doing anything illegal. I appreciate all the help.