Hi Lacey - Welcome to Travelblog!
You can certainly travel from East to West, or West to East, or any way around for that matter, provided you're either a) planning on acquiring a vehicle (you need to be 25 to rent a car in the US without a daily surcharge) b) willing to fly between locations, though you miss out on everything in between or c) have the patience to navigate the bus/train systems. The US has well developed rail in certain sections of the country (the NE for instance) but elsewhere it can be very difficult to travel between locations without a car or a plane, although there are a few train routes that go coast-to-coast.
It takes a minimum of a week to drive from coast-to-coast. That's if you spend 99% of your time driving and sleeping . You won't get to do much other than view things through the window. But you can see much of the country if you give yourself a month or two. We drove through a good portion of the country in about 10 weeks.
In Texas, I would probably recommend Austin which is known for its music scene. I would recommend you stay away from Tijuana. Mexico itself is safe to travel in, but there have definitely been safety issues for Caucasian females in Tijuana lately (according to my sister, who lives in San Diego - this is second hand info. San Diego, by the way, also perfectly safe). Border towns can be hit or miss - tricky US/Mexican politics.
There is less of a backpacker culture in the US than you'll find in Europe, SE Asia or Australia, certainly. But in larger cities you will find hostels, and they will be full other other backpackers. In smaller cities and towns (the Americana part of America) you will be relegated to hotels or, which is gaining popularity here, Couchsurfing.
It is helpful to get an idea of what types of things you like to see, both to give you information on other places to go, but also to help you narrow down your expectations for the big cities. You could live for years in NYC for instance, and not see everything. So - do you like natural wonders, ecology, museums, history, cultural things, music or food? Silly road-side attractions and Americana kitsch? Certain TV or movie destinations? Maybe you want to see the the famous sights? And of course, you don't want to overplan. But having priorities can be good.
If you wanted to start in the east, you could do NYC, NJ, DC, maybe Ontario CA (Toronto) all easy with trains between them. Take a flight down to Florida, and rent a car. See Miami, New Orleans, maybe drive through Savannah GA in between. Depending on what airline you fly, you might have a lay-over in Chicago which you could extend for a few days, than continue to the West Coast. Again, this part of the country is easier by car - San Fran, LA, San Diego, Vegas. There are lots of ways to do it - and also a lot of interesting country in between.
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