How to move apps on SD-card to bigger SD-card without PC
There are a couple of ways to do it:
For simplicity I prefer to split into categories:

Backup of files (music, documents, videos, any other media files and miscellaneous files)

Backup of installed applications

Other (e.g SMS/MMS attachments etc)


Method to back up Categoty 1 Data
Prerequisites:

Unlimited/suitable data plan

Must have signed to a cloud storage (or have one Google drive will do just fine)


You can easily upload the data to google drive or other similar cloud solutions.
To retrieve it you need to redownload it again.
Method to back up Category 2 data
To preserve the intergrity of all your apps (with thier data), you need additional app, capable of doing this. (Install in internal memory).
A good example is Helium - App Sync and Backup (doesnt require  root). Other similar app is Utimate backup, alternatively browse around playstore to pick one there.
It allows you to choose the apps you want to backup. Again if there is no enough space on internal memory use the cloud solution e.g dropbox or similar.

Method to back up category 3 data
This requires much less space than the 2 above. You can just back up in internal memory as well.
Retreiving data after installing new sd card

All the data that have been backup up can then be downloaded again, using the cloud solution previously used and from internal memory.