when only "いきましょう" is written,this means what chpr1 said. Your example "わたしがいきましょう" is usually answer for question what someone are asked as "だれがいきますか". In this sentence,"いきましょう" and "いきます" means the same.so both of the sentence "わたしがいきます" and "わたしがいきましょう" is grammatically correct.Their are more polite than "わたしがいく". If "わたしがいきましょう" is written,I think that old person said this.

Oh yeah, don't memorize "Watashi ga ikimashyou." It is weird.

It means "Let's go!" Use it when you set-out for somewhere. Don't use it when you're going someplace alone. People will think you are crazy.