From the point of view of a bilingual - Mexican American - who speaks Spanish pretty well for a pocho (Americanized Latino), but not as well as most native Latinos - these are some offensively general clues to accent.
Mexicans: sound “normal” to my ears so its hard for me to describe. Some accents are “regional” sounding though. To others may be either “sing song” (northern Mexico ) or “choppy” (central or south Mexico). Also the Ch may have sh sound in north…
Caribbean Islands and Coasts: Speak rather fast, lightly pronounce or “swallow” the s before consonants.
Castillian Spaniards: Lisp the soft c’s and z’s, “mushy” s sound, very distinct intonation (more choppy than smooth). Very strong j’s (more kh sound than h).
Andalusian Spaniards - not as much lisping, fast, similar to Caribbean in intonation.
Argentines, Uruguayans: Very stacatto (perhaps Italian?) intonation, ll and y has zh sound.
Chileans - Sort of inbetween the Argentine and Caribbean.
Andean (more or less from Colombia to Peru). Can be quite like Castillian in intonation, but no lisping. Somewhat fast.
Another big clue is vocabulary. Mexicans use many words of Nahua origin, especially for animals, birds, foods, and so on (zoplilote = buitre, tecolote - bujo, cacahuate = maní…cuate = gémelo ) . These Nahua loan words are often instant give aways that the speaker is Mexican; except for those that have spread elsewhere (tomate, chocolate).
Caribbeans may use some Arawak, Andeans Quechua, etc. as well instead to standard Spanish.