Recommended wineries in Central Rome.
In the center there are many enoteche where you can drink and/or eat. Here are a few:
Casa Bleve: near Piazza Navona. A great place to stop by for lunch, they have an extensive cellar.
Angolo Divino: near Campo di Fiori. Massimo, the owner, is well known for his list of natural and biodynamic wines.
Barrique: in Monti. Interesting wine list, especially his bubblies.
Beppe e i Suoi Formaggi: Ghetto. Not an extensive list, but interesting wines from Piedmonte, paired with the best cheese in town.
Vinoroma organizes formal wine tastings, if you want something like that?
Italy Hotline organizes a few different wine walks in Rome:
Hope this helps!
I think you are looking for recommendations of "enotecas." (Italian for wine bars)
To the best of my knowledge, no wineries have tasting rooms in Rome offering samples of their wines.
Trimani is a well know wine shop that operates a wine bar next door.
Cavour 313 is also a well-established wine bar.
Here is a blog discussing wine bars in Rome: