The tones in Mandarin are considered one of the main problems for Westerners, who come to learn the language, and they have certainly caused me some problems. I think it is hard for someone who has grown up with a tonal language to understand the difficulties of those who have not.
I can say that I have made considerable progress. I try to do little things all the time to improve my feeling for the tones. Initially I was in the same boat as most beginners seem to be, I could hear the tonal difference of syllables but instantly forgot them (I am too used to this sort of audio information being used to carry emotional content in sentences, not changing the meaning of sounds). I also know for a fact that I am still a long way away from the natural ability of a native speaker.
Now I can usually discern tones in clearly spoken Chinese and can often reproduce tones myself. There are difficulties in sentences with some sounds and I have to spend a little time getting used to a new word (in isolation the word it usually easy but in a sentence it is often not so easy). I also have a few blind spots with common words, for example I can say ying1wen2 in isolation but often pronounce it ying2wen2 in the middle of a sentence.
Rather than going crazy over tones I tend to work on them a little all the time. I am constantly attentive to them, and try to do little things to improve. For example if using an online dictionary to check a word I think I have heard, I commit to adding the tone marks when I add enter pinyin (getting these wrong and wasting more time, tends to focus the mind). I sometime concentrate really hard on identifying the tones in Chinese I listen to, even if the words are obvious.
One day, I think I will achieve something close to the natural ability with tones that a native speaker has, I just don't know when. I don't think there is a magic bullet for most of us, just a gradual development. Initially you don't know enough words to get them mixed up anyway (it was quite a revelation when I first mistook lian4xi2 for lian2xi4 when the context was confusing, of course I was also focused on trying to understand the other words).
Some people equate musical ability with tonal language ability, I am not so sure, I can easily tune a guitar by ear, but only to itself, I don't have perfect pitch. I think a general continual attentiveness and time is all that is required. I think even a skilled musician will have problems the first time they encounter a tonal language. We are all different though, I would be interested to hear of other peoples experiences.
I was trying to think of something that may cause Chinese speakers a similar problem when they learn English. Maybe some of the more subtle sentence inflections, for example the hidden 'but'. When a statement is made in English with a slight inflection that tells a native speaker that the word 'but' should be added. "I like what you have done (but)." maybe the sentence that follows takes a non-native speaker by surprise in some cases if they take the first statement at face value?