Hi, you can add fields in the properties, in ArcCatlaog. Alternatively, if you're game, make a copy of your shapefile, then open the .dbf table and save it as an Excel spreadsheet. Cut and paste and add what you need to your table, making sure you adjust the field types in Excel and save as Excel. then Save as .dbf using the same name as the dbf table in your shapefile and replace the original one. If this doesn't work, you can always go back to your copy.
I wouldn't recommend editing .dbf outside ARCGIS, I used to do this in my last job and it corrupted data in many ways, we used to sort the dbf from A-Z only to find after that this changed the ARCGIS shapefile. It can be tedious work but I would actually go for the ArcCatalog method.
You can open the .dbf file usinf Excel, make changes & save as an excel file (.xls) Then open Microsoft Access. In access, u can import an excel sheet.. once this is done use export option & export as .dbf file.
Replace your original .dbf file, with the newly created .dbf file...
An easier solution is to install and use the spreadsheet program in Libre Office or Open Office for dbfs. Both are free open source office suites.
One should never sort or mess with any shape or index field when editing a dbf of a shapefile. We make copies of the shapefile, and edit the dbf in one of them and test the result before making it the current one. Excel is not a good way to add text fields, however, unless you don't mind having them 254 wide. We add the field first in ArcMap and then populate the fields in Excel.
As tonylocandro mentioned, it is not recommended or advised to edit a dbf of a shapefile outside of ArcMap, as it is easy to corrupt data. As for saving an Excel sheet as a dbf, in the Catalog window in ArcMap or ArcCatalog, expand the excel sheet to bring up the individual pages. Then, right click on it > Export > to dBase (single). This will allow you to save the Excel sheet as a dbf.