Yes, both plants and bacterial biofilters will consume fish waste, and keep it from building up. But it's best to have *both* functioning together, each capable of taking up the slack from the other.
With plants alone, depending on how many fish there are, they may not be capable of consuming 100% of fish waste. There may be too few plants for the number of fish. Even if there's technically enough plants, if they're not healthy they won't be effective consumers of fish waste, which can happen if:
1) They're lacking nutrients which the fish produce in insufficient quantities. Typically at least potassium, plus iron and other traces. These must be provided from ferts, or a rich substrate.
2) They're not getting enough light. Or too much light. Or too little carbon for the amount of light.
3) They're in transplant shock from having just been added, from a source with major differences in tank parameters.
4) Or any number of other things.
It can be done, but to guarantee success, it takes a certain amount of experience on behalf of the hobbyist. Since you don't yet have ferts, I think I can safely assume you don't yet have that level of experience, and can't recommend that method. Plants will help, but don't rely on them alone.
Though I've never done it, I hear a fishless cycle can be accomplished with nothing more than fish food. Add the amount you expect to feed daily. Monitor the progress of the cycle with your test kits as normal, and when complete, add the fish. If you've underestimated the amount of food, increase it only *gradually*, so that you don't include a noticeable mini-cycle. This probably means you can't let your children feed, or perhaps even have access to, the fish food without supervision! They'll probably want to "fatten up" the new fish, which could be disastrous.
If you can move a small portion of filter media from one of your established tanks to the new tank, it will greatly speed up the cycle.
Life happens. On semi-hiatus, checking in infrequently.