I find it easy and fun to learn new languages! The only reason I'm any good at programming is that I've got a strong inclination toward language. All human languages are fundamentally the same, and not even vast differences in phonology, grammar, and vocabulary can get in the way of the fact that all people work in basically the same way.
I find it immensely rewarding to draw parallels between etymologies, to discover the underlying nature of what makes a language tick, and to learn how native speakers understand their own language. Not to mention that studying a wide variety of orthographies has given me great clerical accuracy, which is a big help in programming.
However, your mileage may vary—I'm a programmer because I'm a linguist, not the other way round, and you can become proficient at programming in many different ways.
Here are a few tips that I think can help programmers with language learning:
Natural languages are not programming languages. Natural languages do not have rules, but they do have patterns. If you notice a pattern, don't claim it's a rule and then complain about all of the exceptions. Linguistics is not a black-and-white field. I've noticed that people of a technical mindset get caught up in whether they're "correct" and lose sight of the fact that it's more important to be understood. Natural speech has inherent meaning that transcends literalism.
Learning a language is not about rote memorisation. No native speaker of Spanish says to himself "voy, vas, va, vamos, vais, van" to remember how to conjugate "to go". He just does it in running speech because he has developed a sense over time of what sounds right. Do not take a "phrasebook" approach to language learning: you will find yourself lost for the appropriate phrase because you won't be able to produce your own. Learning vocabulary is not the same as learning an API.
Natural languages are redundant and compressible, and you can use this to your advantage as a student. If you pronounce or spell something wrong, chances are you will still be understood. Look up the etymologies of words to get a sense of their deeper meaning. Having a sense of the language as it was is just as important as knowing the language as it is. It's okay to make some mistakes.
Step outside your comfort zone and experiment. Try to talk the way native speakers do. If you notice that you pronounce or articulate something differently, try to discern exactly how. If you don't understand everything someone says, it's okay to ask them to repeat themselves or explain. If you make a mistake, the worst that can happen is a misunderstanding, and if you're confident and outgoing then it turns into a funny situation rather than an awkward, embarrassing one. Have fun.