That's a normal occurrence.
If you are moving to a bigger home, you can either a) sell this home first or b) keep this home and convert it to a rental property.
Most people sell the old home because they don't want to be land lords. But if you elect to keep this current home or are unable to sell it, then you can certainly hang on to it.
To get a mortgage on a new home (if you're keeping the current home), the lender will require you to be able to qualify for both your current mortgage payment AND the new one. In addition they will require you to have money in reserve outside of what is needed to close on the purchase. This will be between 2 and 6 months of payments on both houses, depending on how much equity you have at that time in the current home.
From a rate perspective, you can classify the new home as a primary residence so you won't be charged the higher "investment property" rate on the new loan. But the rates will be whatever they are at that time, which probably won't be as low as they are today.
Hope this helps!