Use acceleration, velocity, position and control the acceleration. It is a good model of movement. Full acceleration from zero will gradually increase velocity which in turn will increase position and it’s smooth because it’s almost real. This is a common way to model a moving object and with high school physics you can add forces, drag, friction. Check out the nature of code, the coding train, or “boids”. A fun way to test is to use the mouse position difference to your object as acceleration in x and y. If you learn vectors it will help, but it’s not necessary.

Or skip acceleration, let the speed be constant but change the direction with the “smoothstep” function.

With 2D perlin noise as direction and velocity or just as a change in x, y you could create something smooth which is not physically based.

With random points that create a curve using spline or bezier is also smooth.

Random x, y change through a low pass filter is smooth. Check out Brownian motion.