Well, I have to say I don't know much Spanish speaking music outside Argentina, just a few songs by a Mexican band named Caifanes (what I heard from them is definitely recommendable, but I haven't listened to any full albums yet) and a lot of folk music and singer-songwriter stuff (Silvio Rodríguez, Violeta Parra, Víctor Jara) where the lyrics play an important role in the songs. I did listen to quite a lot of Brazilian music I could recommend to you, although it's mostly the popular stuff: Caetano Veloso (good entry points are Caetano veloso (1968) ( https://www.youtube.com/playlist?list=PLVnkoLiLMTm6RIArBGspMGPN7rCvnMlhm), Cinema Transcendental ( https://www.youtube.com/playlist?list=PLVnkoLiLMTm5z05I7mwrTNB86kPDz0H2w) and Livro ( https://www.youtube.com/playlist?list=PLrt7VbxNS8recz35ymWv6FlPvYdzlPMIG), but he's rather consistent throughout), Chico Buarque (rather consistent as well, although Construcao and the 4 volumes of Chico Buarque de Hollanda are possibly the best), Gilberto Gil (pretty much everything he released before the late 70s and after the 80s is worth listening to), Tribalistas, early Gal Costa and scattered albums by Joao Gilberto, Clara Nunes, Elis Regina, Vinicius de Moraes and Toquinho.

When it comes to Argentinian music, the first people to get acquainted with are Charly García, Luis Alberto Spinetta and Gustavo Cerati. All of them seemed to prefer writing complicated but still memorable melodies using untrivial chord progressions and several melodic phrases in each section with little repetition. Charly García was my favorite of these (I say "was" because he doesn't really write new music, although he's still alive). He actually had three different bands on the 70s (the first one being the folky/sometimes-proggy Sui Géneris, and the other two being the much proggier La Máquina de Hacer Pájaros, and Serú Girán), before starting a solo career in the 80s, heavily influenced by New Wave and synth-pop. Good entry points for each era could be: Pequeñas anécdotas sobre las instituciones ( https://www.youtube.com/playlist?list=PLWVo2tank-zxIUgqDjKA3EGlz4gtnj5nS), by Sui Géneris; both albums by La Máquina de Hacer Pájaros ( https://www.youtube.com/playlist?list=PL3TpNQfGiuFKvwIy7LSHdV7yxzvkxHWm3;
https://www.youtube.com/playlist?list=PLE0tyPRbYhmL75rago8huoMUZQEnJOLUn); La grasa de las capitales ( https://www.youtube.com/playlist?list=PLIanVw9EBD2GJ0HH8IH68iF0w9ryJFl8m) and the live album No llores por mí, Argentina ( https://youtu.be/U6Uz4XFW6YY), by Serú Girán; and clics modernos ( https://www.youtube.com/playlist?list=PLFHELq0ZTyEG_P4feLxbvu8Qk0HNu9gyY) and Piano Bar ( https://www.youtube.com/playlist?list=PLFHELq0ZTyEFTrkAc6TPoUDYGX5TN3RiS), Garcia's solo career. I do recommend all of it, starting with Sui Generis's first album ( https://www.youtube.com/playlist?list=PLoBhP9ACmehbpQ_Jd7wWVkQi4DkPFt_97) and ending with the artsy concept album La hija de la lágrima ( https://www.youtube.com/playlist?list=PLtijvgFZYZgJQjOWRCEGSWZCR4bWltHM-), the last great album he did, in my opinion (actually, if you don't mind avantgarde concept albums, you might ñoke the follow-up, Say No More). Unfortunately, he got rather lazy after that, mostly sticking to covers and rerecordings.
By the way, if you end up liking Charly Garcia's 80s stuff, and you don't mind high-pitched vocals, you might also like early Fito Páez. He also had a knack for writing really complex but still memorable melodies and giving his songs unusual arrangements and production choices, and some of his earlier songs are classics of Argentinian music. Everything he did before the album Circo Beat (1994) is recommendable (except maybe La La La, which he did with Spinetta during a time when Spinetta was at his most inaccessible), as well as the live album Euforia. Unfortunately, he pretty much lost his songwriting abilities after the 90s.

Luis Alberto Spinetta, like Charly Garcia, had several bands throughout his career. However, he was more experimental, which led to his career being more hit-or-miss than Charly's. In fact, his solo career is an acquired taste for me (I'd only recommend it if you really, really like incredibly complicated melodies, jazz fusion arrangements and 80s production). The stuff he did with his earlier bands, however, is terrific. His first band was Almendra, one of the first Argentinian rock bands, whose first album ( https://www.youtube.com/playlist?list=PL20WkbkmqaonOXVeIKg6vyzLLpv8M8tf0) was a very important one for the development of Argentinian music (the second album does take some time to get used to). His next two bands, Pescado Rabioso and Invisible, are better in my opinion, and they only released three albums each, so you could work your way chronologically.

Gustavo Cerati, then, was the frontman of Soda Stereo, easily the most popular pop band in Argentina. They released a lot of very popular and accessible albums in the 80s, with the distinction that their songs did have typically 80s production, but they usually used a lot of funky guitar patterns and not that many keyboards. Signos ( https://www.youtube.com/playlist?list=PLsa6fFCUllDC7W3iff5Q-aMZz45WAGxjd) is a good entry point into this era of the band.
Then, in the 90s, they switched to shoegaze ( https://www.youtube.com/playlist?list=PLsa6fFCUllDCcQ6mB8K9SaYCEjJEtdia9) and art-pop ( https://www.youtube.com/playlist?list=PLsa6fFCUllDAKbUL_W7kM9T0Ktcs7b-kp) before breaking up. Even though their studio albums are good, I'd say that live is the way to go with them, since they got rid of the sometimes dated production. This live DVD ( https://www.youtube.com/playlist?list=PLrEX0oYRnMW4rZnRfJzOygq49vnmLi3v8) is probably the best way to get the full picture. Gustavo Cerati's solo career is also highly recommendable. He basically continued in the same art-pop direction as on the last Soda Stereo album, experimenting with electronic instruments, samples and intricate production, although the albums are quite different from one another. Particular highlights are the dream-poppy Amor Amarillo ( https://www.youtube.com/playlist?list=PLfUV806q_Ri570YCciKRIcUCGgmzumymX), the very heavily sample based Bocanada ( https://www.youtube.com/playlist?list=PLfUV806q_Ri4F7vDuwkSVLQRU0VwaOemu), and the more hard-rocking Ahí vamos ( https://www.youtube.com/playlist?list=PLsa6fFCUllDBAKqcONd-_VhzNoHAYMF_9).

Mercedes Sosa, one of the singers I linked to you the last time, is a pretty good gateway into Argentinian folk. She didn't write any songs as far as I know, but she was one of the finest performers of the genre. Her albums from the 60s and 70s are nearly all very good to great, although one of my favorites was released in 1996 (Escondido en mi país). I couldn't find many albums on YouTube, unfortunately.
Here are some albums I could find there ( https://www.youtube.com/playlist?list=PL8D35F8E6893AF106; https://www.youtube.com/playlist?list=PL7xhze77UipuSfrZ42Jlk0g4uNg1JVvK4; https://www.youtube.com/playlist?list=PL7xhze77UipubMaHxw_fmCpXxu3m6nm2j; https://www.youtube.com/playlist?list=PLB048DC67281B0C2D). Another important folk artist is León Gieco, who, in his early albums, wrote some of the most memorable Argentinian folk melodies. What he did up to, and including, Pensar en Nada ( https://www.youtube.com/playlist?list=PL426D19B42FC35FAC) is really good, especially El fantasma de Canterville (1976), 4to LP ( https://www.youtube.com/playlist?list=PL3141ED975C4B5FD2) and the aforementioned Pensar en Nada. His late period also includes some of his best music, specifically Bandidos Rurales ( https://www.youtube.com/playlist?list=PL7xhze77Uipu6Tc8Wq7bUeN8kc_KC2O2k) and Orozco ( https://www.youtube.com/playlist?list=PL7xhze77Uipvy9lqREmNMyg75GvYXCOM_). During this time he also released an album of acoustic rerecordings named Desenchufado ( https://www.youtube.com/playlist?list=PL7xhze77Uipu9IAHB1qrxL76aFtEC5GLz) that could serve as a good entry point into his earlier stuff.

For more rocking stuff, there's Divididos, a really cool hard rock band that is notable for their diversity (they often incorporate elements of funk, punk, reggae and folk into their songs), cryptic lyrics, energetic sound and impressive musicianship. I can't guarantee you'll enjoy them, though, because their melodies are not particularly complex (although they do tend to use very unusual chord progressions, and the basslines are often really good), but they certainly are catchy. Good entry points could be La era de la boludez (the most diverse one), Gol de mujer (the most rocking one) and Amapola del 66 ( https://youtu.be/v-r2fDBbB1o) (the moodiest one, and also the one with the most complex melodies). They are also very good live ( https://www.youtube.com/playlist?list=PLgZTCy8VAivhC5FvoxXoCM1Et9NF3c0Sk).

If you don't mind repetition and groove based songs that much, Sumo and Patricio Rey y sus Redonditos de Ricota are also very good rock bands. Sumo had only three official albums and were more of a new wave and reggae influenced band, but at least two of them are considered new wave classics here (Divididos por la felicidad ( https://www.youtube.com/playlist?list=PLQjqJ7yoMDJ41e0UOlW2sLHp23V4HdDSb) and Llegando los monos ( https://www.youtube.com/playlist?list=PLQjqJ7yoMDJ6dOborZBG7rdtcWP1D3bMH)). Patricio Rey y sus Redonditos de Ricota were more conventional, but they were also very moody, and their guitar and sax riffs are some of the catchiest I know. Every album of theirs except the last three is both recommendable and a good entry point. Out of the last three, I'd say Luzbelito is really good, but not exactly representative (by this point they started to rely on more intricate production and eastern motives).

Have fun. :)