It's been around three months since I got a full time position within a company. How long should I wait to?

Check your employee handbook. 9 times out of 10 the handbook has something related to promoting from within. Ask your HR department. Also check the job description of the new position you want. Were you qualified for it when you first started your current position? If not, think about why...this would normally be due to not having enough experience. Also there is a learning curve for any job. Most people claim the learning curve is 2 years to completely learn the job you have. I would say to start doing the things mentioned in the job description for the new position. In this way you can at least say you are already performing those duties.

No need to wait - its your life - go ahead and apply for better position with another employer - if prospective employer wants to talk with your present employer - you must come up with a great way to prevent that! You don't want to tip off your present employer that you are looking around for a better job!