The Godzilla movies are so different that it's hard to make recommendations.
First of all, you should know that they are divided into three series:
Showa (1954 – 1975), Heisei (1984 – 1995) and Milennium (1999 – 2004).
The Showa series starts out fine, but tends to become more kiddie oriented as time goes by. The Heisei series starts with Return of Godzilla (aka Godzilla 1985) which is a sequel to the 1954 Godzilla, ignoring the rest of the Showa series. Then the Milennium series does pretty much the same thing.
The important thing, in my opinion, is that if you chose one from the Showa series, don't chose one directed by Jun Fukuda. Chose one directed by Ishiro Honda.
Destroy All Monsters is fun, but I felt that it was a bit of “throw everything in” at the expense of the story. My favorite besides the original is Mothra vs. Godzilla (1964). Note though that Mothra returns to battle Godzilla in 1992's Godzilla vs. Mothra (aka Godzilla and Mothra: The Battle for Earth).
Also, don't forget that there are some fun non-Godzilla kaijus from Toho, like Mothra or Rodan, or why not Frankenstein Conquers the World. They may not be available on blu-ray yet, though.
