Here are a few things to think about (in addition to becoming one with the notion that the natural sense of が is developed over many years of immersion).
1) I forget where I read this, but I will try to dig up the reference (i have a
lot of Japanese language books, never learned a thing until I found JPod101). 私
は 何何 verb is a natural sentence: "I something something verb". It puts the emphasis of the sentence on the subject/speaker (watashi). 私
が puts the emphasis on
what follows が. For example, 私は高橋です。 I am Takahashi. However, consider the situation where three people who don't know each other very well meet up on the street. Person "A" is Watanabe. Person "B" is Yamada. Person "C" is Takahashi. Watanabe says to Takahashi: 山田さん、お元気ですか?(Yamada-san how are you?, i.e. he mixes up to the two people). Takahashi-san might say: 私
が高橋です。It puts the emphasis on the end of the sentence: I am
Takahashi.
2) が can be used to ask for more information. 新宿に行きたいんですが。I want to go to Shinjuku and... The use of が at the end of the sentence implies that the speaker wants sometihng more. Consider the case where you are a tourist and you are at the concierge desk in the hotel. The natural implication is that you want to know how to get to Shinjuku. Again, note how the emphasis is on something that follows the が which in this case is unsaid: can I get directions? do you have a map? will you take me there?
3) another "more information" example. One of the beginner season 2 lessons (the one about the pub) has: キリンとハイナケンとサッポロがありますが。This is an interesting sentence because it uses が
twice! The first が is the familiar がある form: We
have 何何。(we have these 3 beers). The following が can be thought of as "and..." (in this case the implication is "and, which one do you want"?).
4) This is my favorite. You kind of have to be in Japan to really experience this. You are at a restaurant with friends. You want to pay and you know your friends don't want you to pay (but they make less money than you). The bill comes. You take the bill, they want the bill, they are grabbing for their wallets. You simply say: 私が。That's it, that's the whole sentence. Everyone in Japan above the age of 12 or so will understand that this means "I will pay". This is my favorite example of が。 No way that is a gramatically complete sentence but everyone will understand it in its context. It simply means "I will...". Of course, what follows is "pay".
As others have said don't get too hung up on this. Listen for the natural usages of が and learn them. Collect correct sentences. It's the best way to learn Japanese. Even if some of those sentences don't seem to be complete.