Right, let's see if I may help:
First thing you hear in an English speaker is that they speak too fast. Meaning Spanish stresses every single syllable, while English strikes stress once-twice in a sentence. Speak carefully, take your time. You won't sound limey.
Vowels are very different, I think they are key. English simplifies vowels in unstressed syllables to the so-called schwa. Never do that in Spanish. Only five vowels, but pronounce them clearly and throughly. No pairs long-short, like sheep/ship.
It is a good comparison for the single vibrant r (verde) to put it next to the American inter-vowel t (water, what a day). I agree with you, Rusty, it's no the same, but it is a good pedagogical start for students willing to practice, as it is the case. Similarities are more than differences, I think. If I'm wrong, well, foot in the mouth.
The multiple vibrant r (rabo) is just a repeated variation, as if you imitate the sound of a motorbike.
Another consonat that variates is the t, which in English is slightly africate, at least in European English, while in Spanish it is purely dental.
Hope this helped,
PH