what are other idioms one can use to express getting on well with someone?

I've never heard "like a house on fire," so maybe I don't quite know what you mean. That said, a common idiom for quickly coming to a good relationship with someone would be "hit it off," as in "Joe and Martha just met tonight, but they really seem to be hitting it off."

[Edit]: Some others came to mind for me. Two people who are inseparable might be described as "attached at the hip." Also, two close individuals, particularly if they are similar in many ways, may be called "two peas in a pod."

The Cassell Dictionary of Cliches (1996) confirms that "get along like a house on fire" means "to get on together extremely well." A shorter (and less flamboyant) expression of the same idea is "They get along famously [or swimmingly]." Yet another is "They're hand in glove," signifying a close fit or match of interests between two people. Or you could say that they are "birds of a feather."

Another possibility is to say that the two people "take to each other like a duck to water" or that they "go together like a horse and carriage" (or like bread and butter, or peanut butter and jelly, or salt and pepper, or milk and cookies, or macaroni and cheese, or spaghetti and meatballs, or any complementary pair of things, really).

A less complimentary term for closeness is "thick as thieves."