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As I said, the car has trouble shifting from neutral into first, refusing to shift until I rev the engine to around 3,000rpm.

I'm thinking it's probably the shift solenoid, in which case this:
http://www.1aauto.com/1A/Drivetrain/Dodge/Avenger/1ATRX00005/681932?utm_source=google_base&utm_medium=cpc&utm_campaign=google_base

^ is probably the right part. Do you think I'm right, or should I take it in to the shop before spending my money on the solenoid?  

As a follow-up, I'm an amateur at engine repairs, should I try to do this myself or take it in to the shop? There's no need to pull the tranny.