Here it's much simpler for me, I have some rabbinic guidance. When my mother asked her then-rabbi Adam Mintz (formerly of Lincoln Square, now has his own breakaway minyan - how can the shul rav have a breakaway minyan? it's a long story) what could be chopped to make the Yomim Noroim services shorter at their summer C-nagogue, he said that the most important stuff was baruch she'amar, ashrei, and yishtabach, and on Shabbat or Yom Tov, Nishmat.
When I'm in a hurry, I personally leave in a few other items: Ps. 148 and 150, which most commentators agree are the most important of that sequence of psalms (145-150), Yehi Cavod as a mood-setter for Ashrei, Vayosha so I can praise God for the Exodus.
Some notes on why we say various parts:
Baruch She'amar: a meditation on the Divine Name. Each of the Baruch phrases reflects a different implication of the Divine Name; the last, Baruch Shmo, blesses the Name itself without regard to meaning. It's good to say in that vein before praising God with His Name.
Hodu - see Ta-Shma's Hatefillah He'ashkenazit Hakedumah for a fascinating article suggesting that Hodu may be the oldest continuously-recited human-authored prayer in our liturgy. David Hamelech ordered it written for the Mishkan at Shiloh; it was said in the First Temple before the morning offerings; when Zerubavel was trying to get things started in the Second Temple, he asked the priests what they thought they should say, and they said "we used to thank God" - in other words, say Hodu; it shows up in the Geniza documents as part of the nusach of the Land of Israel (it was not in the Babylonian nussach until much later, when it was absorbed from Ashkenaz along with the Kabbalistic portions), and it's still part of our davening.
OTOH, it's long and a bit confusing towards the end, so I do often skip it.
Yehi Cavod: intro to Ashrei, speaking of God's might and connectedness to us.
Ashrei: the Gemara says that one who says it (with kavvanah) three times daily is assured of the World to Come. Mumbling it doesn't help. It's a tremendous paean to God. The beginning is David's personal praise of God, then it switches over to everyone's praise, then switching at "vechasidecha yevarchucha" to those who are particularly ehrlich - they speak of the glory of His Kingdom, etc. Then at the end it switches back to David.
Ashrei through Ps. 150: there's a Gemara that speaks of the value of "completing the book of Psalms every day", which we short-circuit by just saying these last six. So after Ashrei, one ought to at least say 150 and another one to get whatever merit there is in "finishing" Psalms.
Long historical runup to Az Yashir and Az Yashir: Az Yashir is the archetypal spontaneous praise of God upon salvation, it's the basis for our saying Hallel, so we say it in the morning. But to place it in context, we have the long historical runup. As I said, I often skip this.
Yishtabach: the closing match to Baruch She'amar. It is a bracha on its own, but it does nicely delineate before the Shma brachot.
I do tend to say everything from the Shma brachot through Shmoneh Esreh, then skip around the tachanun and kedusha d'sedra stuff if I'm davening by myself. The Shma brachot are fundamental to the whole experience of Shma, submission to God, confidence in His help, etc. that are the central themes of Shma and Shmoneh Esreh.
To sum up, then, what I do as a minimal davening:
torah brachot, the short brachot, baruch sheamar, yehi cavod, ashrei, ps. 148, ps. 150, vayosha, yishtabach. Everything through Shmoneh Esreh. Aleinu. Shir shel yom.
Some extra tehillim and Nishmat on Shabbat.
As always, YMMV.