To make sure if you can use अपना (or one of its variants, अपने and अपनी) in a sentence, or more precisely, a clause, place अपना where you want to, and then check if it comes after the subject (or all the subjects, if there are multiple subjects) of the clause. If it does, you can use अपना; if it doesn't, you can't. Let me now explain this in the context of the given sentence.
The sentence "नेहा और (उसके/अपने) दोस्त जा रहे हैं" contains two subjects (नेहा and दोस्त), and only one is preceded by अपने. Hence, अपने cannot be used.
But we can modify the sentence a bit so as to have just one subject: नेहा X दोस्तों के साथ जा रही है। (Neha is going with her friends). X is where we are going to see if अपने works. Since the sole subject of the sentence is नेहा, and since it comes before X, अपने can be used.
Glad to know it was helpful. That's definitely one way of looking at it, but it may not always work. I gave some more thought to this topic after posting my first comment and decided to phrase things in a slightly different way this time.
Basically, अपना has to be outside of what constitutes the noun phrase of the clause; having conjunctions inside the noun phrase as long as long as that condition is met shouldn't be a problem.
So, in नेहा और अपने दोस्त जा रहे हैं (noun phrase in italics), अपने is part of the noun phrase, and hence the construction isn't correct.
However, a sentence like नेहा और उसके दोस्त अपने स्कूल जा रहे हैं, (Neha and her friends are going to their school) has "अपने" outside the noun phrase (which also happens to have a conjunction), and hence is perfectly valid.
Finally, one more example with a relatively long noun phrase:
यहाँ रहने वाले और उनके दोस्त कल अपने बच्चों से मिल सकते हैं। - Those who stay here and their friends can meet with their children tomorrow.
दोस्त is actually a masculine noun so its plural is दोस्त itself.
In modern usage, दोस्त is used as if it were feminine when referring to a female friend (उसकी दोस्त). This is iffy grammar but still common enough. However, pluralizing it like a feminine noun (दोस्तें) is a step too far and is not done. You would generally just say उसके दोस्त जा रहे हैं, even if the group of friends was entirely female.
Note: There is another word सहेली which can be used for a female friend of a girl/woman. So, you can also say नेहा और उसकी सहेलियाँ जा रही हैं। for 'Neha and her (female) friends are going'.