Contiki or Solo Travel USA?

I am a kiwi who has lived in London for 2 years and had no problem travelling around Europe on my own but I am a little worried about the USA. I have no car and would be relying on public transport to travel around and from city to city.

I have been told Greyhound buses leave from dodgy areas and that crime in the states is alot more dangerous than europe. Whilst I am sure it is all probably over hypeded hysteria does anyone have any suggestions?

Also are there any budget airlines/coaches/trains or alternatives to contiki people would like to recommend?

You really do end up in some dodgy parts of town when you ride the Greyhounds. Also, they are great for going city to city, but if you are wanting to visit the best parts of America, (the national parks etc.), then they won't really cater for you.

The 'Trek America' and Contiki type tours are excellent, but VERY expensive. Try looking at options with Green Tortoise. ( www.greentortoise.com ).
It's a few years since I travelled with them, but it was a fantastic way to see the country, sleeping either on the bus as you travelled, or around a campfire in the parks.
And it was so much cheaper than the Trek America options.

You're right, it's over hyped. There are, of course, places to avoid but that can be said of anywhere. Greyhound is something I'd avoid. Since I can afford marginally better, I use it.
Anyway, since I'm not a fan of organized tours, my initial suggestion is to organize it yourself. If the Kon Tiki tour takes you everywhere you want to go, by all means, go for it.
Start your planning by deciding what you want to see/experience in the US. As you are aware, it's a huge country with almost limitless options. Maybe head to the library and take out a few USA travel guides and a map. Figure out what you want to see and where you want to go. From there you can organize transport and lodgings. I know you said you have no car so I must assume you have no driving license either. If you do have a license, your options expand immensely.
Traveling around the US alone is fine. I do it all the time. Use your head and be aware of your surroundings, just like you did (or should have been doing) while traveling in europe.
As for budget airlines, there are many regional airlines but the main low-cost carrier is Southwest. I'm a fan of JetBlue. There are more. There are also regional bus (coach) and rail services. Inter-city rail service is provided by Amtrak.

I hope that helped. Enjoy planning! Have a great trip!
If you have more specific questions, feel free to email and I'll answer the best I can.