There's probably no sound so distinctive in American English as the consonant "r". It varies from region to region, being a deep, almost guttural growl in places like New York, a swallowed half vowel in the South, a gentle rear-of-the-mouth sound in the West, and an elided, half-aspirated "h"-like sound in New England.
There is, in other words, no standard "r" sound in American English. It depends where you live, where you grew up, and what if any affectations you want to add to your English. Some people consider a southern accent elegant and a New England accent arrogant, and so adjust their “r” accordingly. The Midwest provides the most standard “r”, and New York City provides the most dominant, obvious one.
So this sound causes no end of difficulty for people learning English in the U.S. It is a difficult sound to learn, even American kids learn it late in the language acquisition process. The problem of the “r”, and mispronouncing “r” and “l” is common enough that linguists call it lallation. Japanese native speakers are quite prone to this, since their language has neither sound; instead having one sound that is somewhere between the two, like a soft “d” made with the tongue touching the region of your mouth above the upper incisors.
The “r” sound is troublesome in other ways. Rhotacism refers to the excessive use of “r” or its mis-articulation as in Bahbwa for Barbara. It is often heard in the southern dialect of American English, and is at times over-used in Cockney English.
In other words, the “r” is a tricky sound, pronounced differently in the languages that have it. Consider the “r” in Spanish, trilled in a way that many Americans struggle to master, or the “r” in French, a gentle sound that comes from the back of the mouth, but sufficiently unusual that few non-natives ever get it quite right.
But back to the American “r”. The above is meant only to show that this sound is troublesome across all languages. If you want to improve your “r” in American English, try the following.
• Practice growling
• Practice snarling
• Pretend you are clearing your throat as you say your "r"
• Imitate the American sound for a dog barking: "ruff-ruff"
Once you are comfortable making these sounds, move on to tongue twisters and wordplay. Here are a few of my favorite examples:
• Reginald Rickles rips rhododendrons rapidly
• The bright red wrinkly wrapper reeks of red pepper
• Release Roderick the rapacious rapscallion
If all this seems like too much trouble, you can also take the Elmer Fudd route: "where's that whaskly wabbit"? Seriously though, the "r" sound will take time, and you may decide you prefer one region's "r" and want to focus on that one. Just practice, and practice, and keep on practicing. That's all it takes.