I always encourage other travelers to focus on pronunciation and getting rid of ones native language accent right from the start. Not to sound more skilled than you are or to hide your origin, but it is necessary to get into the flow and rhythm of a language, which will in turn improve your listening skills and ability to adapt to different accents of that language immensely.
And in geographically widespread languages like spanish or english you have to adapt a lot. As long as you pronounce things in one major regions accent natives have few problems understanding you, even if you make a lot of grammar mistakes. On the other hand, you can have perfect grammar, but with a really thick e.g. english accent, you're sort of limited to talk to people who also speak english and are able to do that head trip.
As a native English speaker who can speak decent Spanish, this was a very interesting experience for me as well. Getting used to saying your own language's words with another language's accent is very instructive. Also hearing English words with a thick Spanish accent was a bit of a head trip for me - at first you don't even recognize them when spoken to you, then after a while you sort of "get" the phonetics enough that you can think of the words as sounding like that.
All in all one of the more interesting aspects of learning to speak a language competently, or alternatively, something that makes the difference between sounding like some American kid in a Spanish class and making a real effort.