There is or there are a table and three chairs

General:
We use the singular form of the verb when the subject is:

(a) singular (b) uncountable (c) a clause

• Sarah is twenty-one today.
  
• The antique furniture in Mrs McDonald’s old house was sold for £14,000.
  
• In good dental care, what is important is regular brushing and a good mouthwash.
  
• What we are interested in is wheat varieties.
  

Threre is OR There are
If the items are being counted separately like the following, :

• There is a table and there are three chairs in the room.
  

Then, you say : There is a table and three chairs in the room.
But Note:

• There are three chairs and a table in the room.
  
  
  
  
  
  

The general rule is that the verb form matches the item(s) that it is adjacent to

adjacent	مجاور
subject	الفاعل