Sorry for your confusion. Me siento bien is the proper form because it comes from the reflexive sentirse (to feel). Siento bien is slang and many of my esteemed cohorts in this forum would say unacceptable. I hear slang used so often I accept it but perhaps it is better for non-native speakers to avoid it.
By the way don't confuse indirect objects (me, te, le) with pronouns such as yo or te. Especially when they are used in reflexive verbs.
Joel
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