The cup looks good. It has a very unique design (the cup is oval and a bit slanted, an exclusive designer cup?). You just need to smooth out some vertices to get rid of the glitches and shading issues.
You can toggle x-ray with Alt+Z. Turn it off to see the mesh better.
Switch to the Sculpt mode (Ctrl+Tab, 2), and smooth the cup with the Smooth brush. Select a low strength (
0.1). You can also activate the Wireframe overlay in Sculpt mode so you can see how the mesh is affected. Touch the bad areas at the bottom and the twisted bottom of the handle. It will lose some volume but don't worry. Just switch to Edit mode and put the vertices back in place.
Disable the Subdivision Surface modifier for Edit mode so you can see the mesh better. Just click on the 2nd icon right next to the modifier's name (the square with the dots, see screenshot). Then you will see the low-poly version of your cup in Edit mode.
Consider reducing the Subdivision Surface to 3 for render. There is no notable difference between 3 and 4. But the number of vertices goes "boom!" if you choose a too-high number.
With G, G you can slide vertices and edge loops to fix the handle.
To make the edge loops of the cup planar, select the loop and press S, Z, 0. That scales it along the Z-axis. The 0 is a special case that tells Blender that you want to have all vertices in the XY-plane (horizontally in a "line"... well, a circle).
Creating the Liquid
If you're happy with the cup you can create the liquid. This is quite simple. Simply select the face inside the bottom of the cup. Activate Face Selection mode (3) and you can select it with one click. Then press Ctrl + Numpad + a few times to select more edge loops (or use menu Select > Select More/Less > More if you don't have a numpad, and Shift+R to repeat the last action).
Then duplicate (D, Esc) and separate the selected mesh (P). Flip the normals (Alt+N), and close the top (insect a few times and and grid fill, for example). This is shown in the tutorial by Blender Guru.