How can I achieve a smooth animation in Adobe Premiere?
You need to add ease-in to the motion. That's where the motion slows as it approaches the final position. You can add it quickly by hitting the f9 key. This sets the speed of the keyframe to zero, and gives it 16% influence. To change the effect you cn edit this by opening the keyframe properties dialogue. Hit ctrl/⌘shiftk. Alternatively open the graph editor and click on the position property to show the graph. You can adjust the speed of the keyframe and its influence by dragging the Bézier handles there.

You might want to duckduckgo the 12 principles of animation, if you want to do some extra reading. This is a valuable guide to creating pleasing motion in animation and motion graphics, that is as true now as it was when it was written back in the days of cels and paint.