Here are some of the most common issues, as well as some words for practicing:
The letter "z" in Spanish is pronounced like the English "s": razón, vez, paz, azul. You will only hear a "zz" sound in Spanish before a voiced consonant: mismo, desde.
The letter "h" makes no sound: hola, Alhambra, alcohol, almohada.
The final vowel should not close. English closes final vowels. Notice what your mouth does at the end of "no" in English. It's as if it were spelled "nou". Spanish final vowels do not close: se, no, tengo, hablo, ve.
“b”, “d” and “g” have softer versions, in which the air never stops flowing. Think of English "about", "ado", and "ago". The air is interrupted at those consonants. In Spanish, those consonants have softer versions whenever they are between two vowels: abogado, abrigada
Generally, "b" and "v" are pronounced with both lips together, albeit often softly. Don't use your lower teeth like the English "v": vivir, cava, mover, verbo
"t" and "d" are pronounced with the tongue on the teeth. In the case of the soft "d" described above, the tongue can even be between the teeth: cada, codo, foto, mito.
[r] is a flap, which is what English speakers do to th e"t" in "water" when they pronounce it naturally: cara, cero, dura, oro. Only roll your r when it begins a word, is spelled with rr, or follows an s, l, or n.
The letter "u" is just one sound. Careful with English cognates that would sound like "iu". In the Spanish version, it's just "u": puro, mutuo, cura
And finally, avoid schwas. English loves schwas. Schwa is that neutral vowel noise that kinda sounds like "uh" in unstressed syllables in words like "about" (the letter "a") and "necessary" (the second letter "e"). Notice if you're throwing schwas into the unstressed syllables in Spanish: pantalla (watch that first "a") pesimista (watch the first "i"), necesitamos (watch the first "e")