The best way is to use a card reader in your PC. Put the new card in the reader and insert it into your PC. Make sure it's formatted as FAT32. With the old card in your phone/tablet, connect it to your PC with a USB cable. Then, copy everything on your old SD card to your new SD card. Make sure that you enable file transfer mode (MTP) and that on your PC you enable "show hidden or system files". When finished, shut off the phone, swap cards and reboot. It should boot up just like it was the old card, just bigger.
One caveat. If your SD card is encrypted or mounted as expandable memory and not simply storage, this won't work. You'll need to use the instructions below.
If your card is encrypted or you don't have an SD card reader, then with the old card in your phone/tablet, connect it to your PC with a USB cable (enable file transfer mode (MTP) on your phone and on your PC you enable "show hidden or system files"). Then, copy everything on your old SD card to a folder on your PC. When it finishes, you eject the phone properly from the PC and shut the phone down. Swap in the new SD card and when it's booted completely, plug it back into your PC and copy those files back. to the new card. The problem with this is that on startup, it will detect the new card and might give you options to encrypt and/or use as expandable memory. There's nothing wrong with either as long as you understand what they are. And, while the phone is booting, it's going to recreate a lot of folders that it can't find on the SD card. Overwriting them with old data, might cause some apps to have problems, so in this case, I'd do it on a folder by folder basis just to be safe.
If all you are copying is media like misc or photos, then either method is fine.