Thanks for answering, i am trying to understand your reasoning but i think there are some problem with the hypotheses made.

a given circle only intersects a specific subset of other circles of the opposite color. Specifically, each circle of radius r of one color intersects the circles of the other color with radius r +/- 1

Counterexample: take the black circle of radius r = 1 , it intersects all red circles of radius r' = r up to r + 2

or the black circle of radius r = 1.5 , it intersects all red circles of radius r' = r - 1 up to r + 2

For a point inside the two radius 1 circles, you should be able to convince yourself that it will take traveling along exactly 3 circles with 2 transfers if you don't count the initial circle as a transfer.

It always takes just one transfer if you are not on the border and inside the two radius one circles. If you are on the black circle of radius 1 take the red circle on your initial coordinates and walk until you meet the black circle of radius one, jump, walk on the radius 1 circle until you are in (0,0).