As human beings, we are far more comfortable when we do things together. So it's a bit counter to our nature to say "I'm going alone." But you CAN go alone, have a fabulous time, stay within a budget, and come back refreshed.
1. Travel within your comfort zone - at first.
If you're from a town with fewer than 1,000 people, and the thought of being in a huge city stresses you out, do research on something you love, find where the heart of it is, and go. Flying into big cities but exploring the suburbs or national parks within driving distance can also give you a little taste of what the city is like - without forcing you to be where you're not comfortable. My best example of this is for wine aficionados. There are some FANTASTIC winery finds around the United States, and while you may have to fly into a bigger airport to get a good flight at a reasonable rate, wine country is nothing if not relaxed and evenly paced. If you're looking for adventure, travel in the summer time when the days are longer and you'll have more daylight hours to explore. When you've got your travel legs under you, branch out to the bigger cities.
2. Give yourself enough time.
If you have a week off or less for your vacation, plan to stay in one place. You can always change your plans as you explore, but if you really want to get to know the character of a city and the locals, really live there for a few days. Go to the grocery store, even if it's just to see what they have that's different. I know in our local independent grocery stores, we have some great Basque foods that are really unique, so you can always get some local flavor to take or ship home. Most importantly, know your travel area and what's reasonable for a day. Don't plan to drive more than 100 miles in any direction unless you're switching hotels. You don't want to spend your entire vacation going from point A to point B and back.
3. Give yourself plenty of YOU time.
When you get to the city, plan half your day, and leave yourself plenty of time for stopping, staring, resting, taking pictures, drinking coffee, or whatever you enjoy. If you're in Sedona, you're going to appreciate having an hour to just stare at the grandeur of the desert southwest's red rock formations. If you're in Manhattan, plan a harbor tour. When you're on a boat and cruising around the island, you really do develop a new perspective for the city itself.
4. Travel light.
One luggage bag. One multi-purpose bag. Period. You can wear clothing more than once (unless you're planning on hiking the Grand Canyon - and then I'd recommend bringing a couple changes of clothes and finding a good laundromat). You don't need your huge can of hairspray, your personal hair dryer, hot rollers, and make up bag. Take your essentials. You'll thank yourself when you're packing to go home, having to change hotels, or end up walking further than you thought you'd have to.
5. Experience the local hotels.
I've stayed in No Star and Four Star hotels, and by far, I've met some of the most pleasant, wonderful people in the No Stars over any hotel that is referred to as "a property." If you find yourself at the hotel when the maid comes by, TALK to him or her (most likely her - I know). Look her in the eye and thank her for the work she is doing. Tip her, right then and there (yes, tipping the maid staff is still appropriate and acceptable). And when you go to get something to eat, go to a hotel bar whether you drink or not or a cafe with counter seating, and talk to the bartender or wait staff if they have time. You'll thank yourself for it.
6. Take public transportation
When you travel light, your options for experiencing local travel open up. You can take a cab or public transportation without needing to corral your luggage. If you have the option to take a subway, train, or bus system, you'll be right in the middle of the culture of the locals. Taking the time to learn how to get around prior to going using the public system will give you numerous opportunities for people watching. And when you're on the bus that goes by the local store, get off and see what people are buying and who is shopping there. You'll be amazed at what you can learn from a culture when you see how they REALLY live - not just what is available at your hotel.