I traveled to Washington D.C. for a few days on business by myself a few years ago. I toured the city alone to visit the sights and like you I wanted some images of myself in front of the famous landmarks.
To achieve this I did a few things
- Mounted the camera on a tripod
- Used a wide angle lens to allow for cropping later
- When possible shot with a smaller aperture for a greater depth of field
- Used the cameras self timer function or remote if you have one
- Looked for surrounding objects in places that I will stand for the shot
One challenge is to accurately lock in the focus while using a tripod. I did have to estimate the distance to focus at, especially for the shot at f/2.8. You can either use the distance window on the higher end Canon lenses to help determine what to focus at, or you can shoot right next to an object that is just out of the frame - focus on that object first, then recompose to remove the object but keep focus the same.
All of these shots below were with a Canon 40D and a Canon 17-55mm f/2.8 IS USM lens, mounted on a lightweight travel tripod that I brought along. The tripod was small enough to throw in a backpack when I went into museums or was not using it.
Frankly, I'm not comfortable handing someone $2,000 in camera equipment, and they will probably do a bad job of using it anyways. A point and shoot is an option, but will obviously not give you the same quality as a DSLR might.
If you don't want to bring a tripod, you can use almost all of the same techniques, you will just have to look out for objects to position the camera on that can act as a tripod. You will also want to consider how much of the image could be of the object the camera is sitting on, that you will later want to crop out.
If all else fails, and the above isn't working for you, find a touristy looking person with an SLR already around their neck, and ask them to take the shot for you. They won't be afraid of your camera, and they probably aren't going to run away and steal someone else's camera anyways.
44mm f/2.8 1/2500sec ISO 250
17mm f/9 1/250sec ISO 200
40mm f/9 1/125sec ISO 200