I started with 'Broken Flowers' because I was a fan of Bill Murray's deadpan humour and 'Lost in Translation' is one of my favourite films. Thankfully, 'Broken Flowers' is also one of Jarmusch's most accessible films and I became an instant fan.
The straightforward and likable 'Down By Law', as well as the two episodic films, 'Mystery Train' and 'Night On Earth', are also easy to enjoy. They have a very cool atmosphere.
While Jarmusch's genre experiments are just as rewarding, it'd probably be best if you knew what you were getting into before you watched any of them. 'Ghost Dog' is an ironic take on gangster films, 'Dead Man' is a very bizarre acid western and 'The Limits of Control' is an allegorical hitman film. I would say these films are love-hate ones. I know people who are fans of Jarmusch who nevertheless dislike one or the other of these three.
If I had to rank the films from most to least accessible for newcomers to Jarmusch, I'd say

1. Broken Flowers
2. Down By Law
3. Mystery Train
4. Night on Earth
5. Ghost Dog: Way of the Samurai
6. Strangers Than Paradise
7. Dead Man
8. Coffee and Cigarettes
9. The Limits of Control
   If I were to rank them in order of personal preference, 'Dead Man' would probably be higher on the list. I love Westerns, I love writers like Cormac McCarthy, and I found 'Dead Man' to be very enjoyable.
   Long story short all Jarmusch is worth your time.

I'd start with Stranger Than Paradise or Down By Law. They are my first two.Jarmusch films, and none of the others so far have disappointed.
Also recommended:
Dead Man
Ghost Dog: Way of the Samurai
Broken Flowers
Night on Earth
Haven't seen (but plan to):
5b4
Mystery Train
The Limits of Control
Only Lovers Left Alive
Coffee and Cigarettes is fun, but mostly a for-fans-only kind of experiment. See the others first.