For 4x4 you should probably save up some money for the magnetic ones like the YuSu v2 M, Yuxin Little Magic M,etc as budget 4x4s are not so good as of now.I had heard that the 4x4 is pretty bad. I'm gonna wait to learn 4x4 until my times are a little better with 3x3. Maybe like under a minute.
One budget cube that I've heard is good is the Meilong 4x4, so that might be an option.
Also consider buying from specialized speedcubing stores such as Speedcubeshop or TheCubicle, as they offer a wider selection of cubes, and sell lubricants too!