Ways of saying "to get into a relationship."

Solution 1:

You can "fire up"/"ignite"/"kindle" a new relationship or re-ignite/re-kindle an old one.

For primarily non-romantic purposes, one can also "establish a relationship."

Edited to add "finding/found oneself/ourselves/themselves" in a relationship:

After years of friendship, we found ourselves in a relationship. The rest is history.

(Not to be confused with: "After years of marriage, we found ourselves in a relationship going nowhere"!)

Solution 2:

To strike up can be appropriate in this situation: see 81c in the provided link. In this sense it means to begin, and is commonly used regarding acquaintance or relationship.

Solution 3:

• Started a relationship - "They felt so close they started a relationship"

• Less formally: Become an item - "They became an item almost as soon as they met"

• More formally: Entered a relationship (not normally used for romantic relationships!)

Solution 4:

You could

• slide
• ease
• tumble
• drift
• leap
• jump

into a relationship. And many more modes of movement, I am sure.