Left-leaning media outlets in the US demonstrate their bias mostly in the following ways: less newsworthy stories favorable to the right are generally not covered; headlines may be slanted to favor the left; talk-shows are obviously left-leaning.
Right-leaning media outlets in the US mirror the biases of left-leaning media, except that coverage favors the right: less newsworthy stores favorable to the left are mostly not covered; headlines are slanted to favor the right; talk-shows are obviously right-leaning.
What differentiates left-leaning outlets from right-leaning ones, is that most right-leaning media outlets no longer stick to the facts, and indeed report known-falsehoods in cases where the truth is unpopular with their audience. Right-leaning talkshows are most affected by this; but on the right, broadcasts purported to be strictly news, sometimes (or oftentimes depending on the outlet), also propagate known falsehoods.
For those interested in the perceived biases of various news outlets in the US, Allsides has a breakdown which may prove of interest:
https://www.allsides.com/media-bias/media-bias-chart.
Edit:
Is anyone else surprised that The Wall Street Journal is classified as centrist in Allside's breakdown? For starters Murdoch owns The Wall Street Journal which almost certainly is going to color their news coverage. And secondly, it strikes me that most Wall Street Journal opinion pieces are right-leaning.