I'll start with learning consonants that do not exist in my own language. These are [ç] as in I[ç] or Tei[ç], [x] as in Ba[x] or Da[x] (Bach/Dach) and especially the German r, which is usually rendered as [ʁ]. There is a variant of [x], which is [χ], the unvoiced counterpart of [ʁ].
The American "r" is a sure telltale sign, by the way.
A sound you already know, [v], is the common way to pronounce "w". The [w] in engl. "water" doesn't exist except for loan words.
"s" has to be rendered as [z] in High German (as in "design") when appearing as initial sound and as [s] when it appears as final sound.
Other sounds change as well, when they appear as final sound:
- b => p
- d => t
- g => k
- [ʁ] => ɐ (for unstressed endings such as -er, -ur)
Another important property is that every German syllable begins with a consonant; however, there is one that doesn't appear in writing: the glottal stop ʔ.
Vowels are much harder to learn. I'd propose to just exercise speaking short words again and again.