How important is it to remember correct genders?

the notion of gender in French is one of the most difficult points to grasp in the language, and especially for the English-speaking world. French distinguishes the masculine from the feminine whether for people as for "objects". there is also the "neutral" gender (who is not male nor female).

below 2 very light description links, but which can be used as a basis. "le vs la" http://www.frenchtutorial.com/en/learn-french/basics/le_la_les
"genders" https://www.france-pub.com/french-masculine-feminine.php in english with french sample

know that in French for example: "une armoire" (a cupboard) on one side and on the other hand "un meuble" (a furniture) on the other, yet a cupboard is a piece of furniture ;) (une armoire est un meuble)

other sample .. "un arbre est dans la forêt" (a tree is in the forest) we say "la forêt" (a designed forest) and une "forêt" (one of forest)

But, "un arbre" (a tree) can't say "le arbre" ! -> "les arbres" or "des arbres" (trees) is ok (

there are no grammatical global rules (except for certain cases), and it will simply be necessary to know if it is an "le/un" or "la/une". this is one of the mistakes (funny for the French) that all non-French speaking foreigners do the most You learn it n time

Here a review on the knowledge of "gender" at school (9 year olds) http://ekladata.com/93tBec2fivol3yOFXLCMpdCCwjo/Fiches-Orthographe-CM1.pdf practice, and get corrected ;)

If the gender is uncertain/wrong, context would be the biggest indicator in a conversation. For example, you've almost definitely heard of the "Tour de France" and the "Eiffel Tower." In French, these are <<Le Tour de France>> and <<La Tour Eiffel>> respectively. Le Tour would be the ride/turn, while La Tour would be the tower.

Regardless, grammatical genders are important in any language using them, unless proper grammar is not an issue.

If I were to learn French but I got the gender of objects wrong, would what I was saying be understandable by a French speaker?

Yes, of course, assuming the rest of your French is correct enough. Using wrong noun genders is a very common mistake made by people for whom French is not the mother tongue, and especially native English speakers for whom this concept is foreign.