Not sure I'd suggest The Door into Summer, it's one of the weakest books from Heinlein imho together with the Number of the Beast and I Will Fear No Evil. I'd usually recommend the Moon is a Harsh Mistress or one of his juveniles as the first book.
I remember reading the serialization of the first few chapters of Number of the Beast in Omni magazine, and thinking how great it was. Then I got the book and read the rest, and was sad.
Don't get me started on _Friday_.
I think part of my problem with Fear no Evil is that I read it just after reading all of the Lazarus Long stories (which I enjoyed a lot) and coming from that Fear no Evil just wasn't as good...
I actually did like Job A Comedy of Justice which he wrote after Fear no Evil... It's very different from his other books and I wouldn't say it's great but it's fun.
Friday is one of his few novels that I never read..