You can right-click on the database in the Object Explorer and do a Task > Generate Scripts.
That allows you to pick a whole bunch of objects to be scripted (e.g. tables, views, stored procs) and you can store those into a single big SQL file, or one SQL file per object. Works really quite well!
Update: if you want to do this in the SQL Server Management Studio app, you can use this SQL script to find the stored procs and their definitions - you cannot however have SQL Server Mgmt Studio write out the files to disk, that doesn't work - but you can copy the results into e.g. Excel.
SELECT
pr.name ,
pr.type_desc ,
pr.create_date ,
mod.definition
FROM sys.procedures pr
INNER JOIN sys.sql_modules mod ON pr.object_id = mod.object_id
WHERE pr.Is_MS_Shipped = 0