For Ratbeard you will get a notice from the character (also on the companion page he will be marked by a !) and speaking to him will trigger a quest-line (No experience, no gold, the only thing you get is a better companion). You do the questline...the first for Ratbeard is a bit of running around and combat and not too bad. You will het another when he hits 28 (he is the first I got to hit second promotion quest) which is more difficult. For me, the longest quest was Old Scratch's 2nd promotion quest at lvl 35 (started in Cool Ranch, to Skull Island, Cool Ranch, Avernus Skyway, Cool Ranch....lets just say a lot of time in the skyways). It was also the most interesting for me...though I am waiting for Dead Mike's 2nd...
To promote any Companion, you have to wait until the Companion says that he/she needs to talk to you in a tavern. You can also see an Exclamation Mark on the Companion's Icon in the top left corner (Companion Management Interface 'u'). The Promotion Quest (my name for it) occurs when your Companion reaches a certain level and most of the Companions see to have their own level for Promotion Requests.
Like all companions, once you get them to a certain level you'll get a notice saying that that specific companion is ready to promote. When they are, all you need to do is go to a tavern cellar, interact with them and go from there. The companions you get via main quests like old scratch and ratbeard will have quests associated with them, while companions not crucial to the storyline like left and ghost singer will have you pay a gold fee in order to promote them.
To be able to help you, you need to give us more detail.
What level is your Ratbeard?
If he is ready for promotion (after you use training tomes), then there is an exclamation point where is face is on the companion list. Ratbeard will also speak to you about going to a tavern (basement), and you need to talk to him there, so you and him will know what to do to get him promoted. The quest for his promotion will be with the rest of your quests, and will have his picture at the bottom of the specific quest box.