Focusing is much more than just point and shoot.
If that is the case then one potential issue is the camera is focusing on something that you don't want to focus on.
I don't know the Canon 60d so I cannot specifically elaborate. But this article may shed some light on it for you ==>
Focusing Quick Tip: Single Point AF Methods
If you can provide examples too we can help more.
If you are taking pictures of something small in the overall scene then the camera may not even notice it to focus on it. For instance, let's say you want to take a photo of your friend while there are many people roving around. How does the camera know who to focus on? It doesn't. Sometimes you have to take control and make that decision for the camera.
There are multiple things on a camera you can control, such as:
- Exposure Mode (Manual which is for Shutter, Aperture and ISO)
- Metering Mode (single point up to entire scene evaluation
- Focus Modes and Areas - the linked article reviews the "areas" part of it to focus on what you want to focus on, not letting the camera make a selection.
Lowering a shutter speed may introduce motion blur not only from the subject but also from the person holding the camera. Don't be afraid of ISO unless your camera is fairly old. The newer cameras are very good at low noise ISO thus allowing more flexibility in higher shutter speeds and aperture control.