To surf those waves, Surfthosewaves you would be OK with buying a fun board of the dimensions of the one you are currently renting .
Does your mom surf? I suspect she thinks a fun board is a toy, like a bicycle with trainer wheels, that label is really only about soft tops and foam boards IMO.
This part is not disrespectful but I think her idea that you will grow out of a funboard soon is wrong
If you get a fun board you will be able to learn and improve your surfing on it for quite a while, fun boards are only a label and plenty of people have ridden their NSPs for a lot of years before they upgrade.
Stick around the range of size you feel comfortable with at the moment practice and get your skills and water knowledge improving and then you will know when you need to change boards. This will also take into account your growth in size and strength.
There should be some NSP boards secondhand around. But new NSPs are not so dear, and if you are going to have it for some years then the price will seem a little better.
By the way you are being really sensible asking rather than just rushing out and buying.
Now about the wetsuit, that is a little more challenging, as you are 12 you are likely to do some growing and the problem with that is that wetsuits need to be a snug fit to be effective there are plenty of surf shops around you will need to try them on but be aware that if you have a growth spurt, they will only stretch so far and you will need a new one.
Show your mom my reply to your post and if she wants to ask me questions I'd be happy to post answers for her.
Hope this helps you begin your surfing journey

I've taken up troll hunting just for fun, instead of a rifle I'll just use a pun! 冲浪爷爷