Falconer,
It sounds like purchasing your first home is a great goal for your first investment. Just remember that a lot of mortgages require that you put 20% down, and in a lot of states that will also prevent you from needing additional mortgage insurance.
Before you invest in a 1 bedroom apartment, keep an eye on comps in the area and see how quickly they sell. Even though it may be a good rental investment later if you outgrow the apartment, you want to make sure you will be able to sell it when you need to.
Also, before you put all of your extra money into paying off your mortgage, make sure you are maxing out other tax advantaged investment options that you apply for such as your employer's 401k, Roth IRA, etc. Right now a lot of people are well exceeding their mortgage rate in investment returns, so it makes sense to invest some of that extra money rather than paying off a low interest mortgage of 3-4% quickly.
Good luck on saving up your first down payment and your first home purchase!