First, welcome to Fish Lore
I assume you have a sponge, activated carbon / charcoal, and some biological media (ex. bio-cubes) in your filter. Whenever you clean your filter, ALWAYS do it in tank water. Cleaning your filter media in chlorinated tap water will kill off the beneficial bacteria. Take some tank water in a separate bucket/container. Then simply take filter media out of your filter and clean them in the bucket/container with your tank water. Take more tank water if the one in the bucket/container gets too dirty.
You should rinse/squeeze the sponge gently (but well enough to remove trapped debris and wastes from it). You don't need to throw the sponge away for as long as it will be simply too old and falling apart to be kept in your filter. It's not true that you need to replace it every month - the manufacturer tell you so only to make money. If you throw your sponge away, you're throwing all your nitrifying bacteria and will have to cycle again.
As for the activated carbon / charcoal, this one has to be replaced every month because it becomes ineffective virtually after only 1 - 2 weeks of use. After a month, it may start leaching the toxins back to the tank. If you don't want to, you don't have to use activated carbon / charcoal in your filter either (provided that your water changing habits are good and your tank is not overstocked). If you decide not to use it, you can put an additional sponge in the place of activated carbon / charcoal - this will give even more space for the beneficial bacteria to grow on.
And the biological media don't have to be ever replaced. They contain vast amounts of nitrifying bacteria. They're there to ensure the cycle continues on even after you've changed one medium such as a sponge for example. Bio-media are not like sponges so they don't get clogged up with debris. You only need to rinse them gently, that's all.
Best of luck with everything
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