You are going to have a very difficult time achieving the first 4 with any band on a low budget. But in general, I'll say a few words to get you started.
- You probably want to be able to use digital modes. Your best bet to get consistently across the country will be via digital modes, as they can add something like 20-30 dB to your signal. Olivia in particular might be a good choice.
- If you want to be able to reach your "neighborhood" consistently, and occasionally around the world, then your best bet is probably 40 m. In fact, 40m is a highly recommended band to start out on. It works very reliably up to about 700 miles during the day, a bit further at night, and if you get it high enough, can work really well at night. Plus it's less crowded and generally more friendly than 20m.
- 20m is the best band overall to work around the world, but you often will have difficulties with things that are closer.
- If you really want voice to anywhere in the world, you'll want a huge yagi antenna on a large tower, transmitting at the full legal power in a quiet RF area. Other than that, you will have some compromise.
My set up is a G5RV in my attic, which I can talk routinely to Europe on 20m, most of the East part of North America on 40m, and occasionally further on 10 and 15m. I live in Virginia (FM19), to give you an idea.