What's the most inexpensive (but effective!) way to fix the gaps between my teeth?

Gaps between teeth: Bonding, but it may not last and can discolor over time. Veneers or invisalign are better options if you can afford it. It may be quicker and cheaper than you thing. Get an evaluation.

Gaps between teeth: Usually exist due to a deep bite, large tongue, tongue thrust habit, small teeth, TMJ problem.., to name a few. Ideally, it is better to find out the etiology why the condition exists to prevent failure of treatment and achieve lasting restoration (if needed on top of ortho treatment). Bonding can work & cheapergap if the bite is within normal limits and not merely caused by teeth size discrepancy.