"Writing" is just a simple way of saying "saving to the SD card. It is just like saving something on your computer. No special tools required. If you want me to give you the long drawn out explaination on how it happens I can, but lets just let it go at what I have said.
It is not so much that it will not write TO a class 10 card, it just may not be able to write at the maxium speed the card will allow.
Lokk at it this way, a Car can go 200 MPH, doesn't mean that it will.
In this day and age, I cannot see you even finding just a card reader, most of them do both right out of the box.
Transferring data is just this easy. Hook your phone (with your old card in it) up to your computer. When it shows up on the computer, move all the files from the card to a folder on your computer. Disconnect the phone, insert the new card, reconnect, move the files back.
Depending on how much data you have it shouldnt take more than 15 minutes or so.
PS there are no dumb or naive-sounding questions here on AC. We are here to help, do not hesitate to ask away.