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.