Hello,
The best way to approach the matter is to add fields by creating custom module(s). This way, you can setup the custom module on your DEV installation, install it and check if it's working properly. Once you get the proper results, you can copy the module on PROD server and simply install the module. Adding new fields through code it pretty straight-forward.
If that's not an option, then there are two tables that you need to take into consideration when moving fields between installation. One is [RecordSet]_data_1 as mentioned by Erwin, the other is [RecordSet]_field.
From the *_data_1 table, you want to copy the columns f_[field name] over to PROD installation. The next step is to copy the contents of *_field over to the PROD server. This time, you either make sure that the DEV has all the fields needed and you can overwrite the whole contents, or you have to manually move each row for all the new fields created.
To be perfectly honest, the 2nd option is much more likely to create issues than simply adding the fields to PROD directly. I would simply recommend creating a backup before every change and it should give you better and faster results.
Kind regards,
Arek