There are different kinds of relationships. Some examples are:
- romantic relationships
- relationships with your family members
- business relationships
When you're talking about starting a business relationship, you use the word "establish":
We're hoping to establish an ongoing relationship with a manufacturer in Ireland.
It's a little strange to use "establish" to talk about starting other kinds of relationships. Use "start" instead:
I'm sorry. I'm just not ready to start a relationship.
A business relationship can be formal, meaning that there's been a specific agreement for two companies to work together. Or it can be informal, which means that companies do business together because it seems like a good idea at the time.
You can also establish a personal relationship with a business contact. That might include getting to know things about their family, their hobbies, and their tastes.