Very new to Illustrator, trying to sort it out, created a few shapes, ellipse and rectangle, and then tried to join them together through Illustrator's Pathfinder > Unite effect and ended up with rough jagged corners where the two meet together. How can I clean this up? or use the same effect without the flaws?
That looks like the shapes weren't meeting properly before you applied the unite.
To ensure that you get a nice clean join you need to make sure that there is a good overlap between the shapes before uniting them. In this case, that would mean extending the square shape upwards so that it protrudes further into the ellipse.