Hey ChickenSlurpee,
This is totally doable in 3ds Max.
One of the areas to look at is the amount of vertices and faces you have for what you are modeling. Every rectangular face is actually two triangles. Your poly count (number of faces) is very important in modeling. So picture a circle; a very low poly count circle will look like an octagon or even a triangle. The more faces you add, the smoother the circle.
Same applies to 3D models; for your mug, you have very large faces. In order to smooth, round, fillet, etc. you need to add faces/polys. So to make the hole in your handle, you need edges for the hole so those faces can be moved, removed or opened to create a hole.
To play with this, make a sphere with 16 divisions. Then run the Optimize modifier and slide it up and down; you'll see your sphere become a diamond, cube, etc.
This is a very oversimplified example, but I wanted you to understand that to take a cube and round the corners means your cube would need to be divided into at least 9 sections per side; the corners on each end plus the smooth middle. If you have less than 9 sections, when you round, your cube becomes spherical. So add vertices and polys to those areas you want to have higher detail.
I hope this helps point you in the direction you need.
-Kenny
