Forget about other people!
Don't be afraid to look silly with a selfie-stick, or if you can master the one-hand selfie with just your phone, just do it. Get yourself in those pictures! I often forget to capture a moment because I am immersed in it. Sometimes, a site is so enamoring I lose myself to it and later think to myself "Why the heck didn't I think to take a picture?"
Eating Out Alone
Want a nice meal but feel silly eating in a restaurant alone? The only way to get over this is to just do it. Bring a book, grab a local paper (usually free in many eateries), or simply play on your phone until dinner arrives. I have had families ask me to join them and I have does this before. It can be great fun and you don't need to do much talking.
If you feel the need to talk you can always ask your new friends about themselves - people love it when others show interest in them. When asked to join others, but do not want to, a simple no thank you usually does the trick. If you're a woman and a man doesn't take "No, Thank you" simply be firm. Don't be afraid to let the staff know you are being bothered if you feel someone is invading your space.
Don't Wait for Others
Want to do something but, again, being solo is awkward? Fairs, festivals, and concerts are fantastic activities where you can go as unnoticed as you want. Maybe you're one of the few lucky enough to have a partner/best friend with the same schedule and exact tastes as you have there will be activities you want to do. If your like the rest of us there will be places you want to go but will miss out on if you wait for someone else to join you.