So I'm a person of Indian origin who has spent half his life in Massachusetts, and a couple of years ago moved to Mumbai.
I personally was always able to switch between accents, but in the past I could not use or understand Indian English to its complete extent.
I agree with @TimPost — you should be fine with asking around — but it is a useful skill if you plan to be here for a while.
Problem #1 - words and phrases used in Indian English that aren't in British English. I've mostly got the hang of lakh, and I can normally guess what things like "please do the needful" mean, but there are quite a few others I'm still not understanding / understanding correctly
This usually can be picked up by just talking to people. Newspapers and the like unfortunately (fortunately?) try to use correct English without any new phrases, though new words like "lakh", "babu", "bandh", "bungalow", "tiffin", "ragging", "ghat" (to name a few) are commonly used.
For what it's worth, most phrases are local — aside from a few universal ones ("do the needful"), most are specific to a locality and you probably will have a hard time finding a compilation of locality-specific phrases.
As for mangled grammar, I've found that there isn't any fixed style to it — while a lot of people tend to pattern their English on the basis of their Indic mother tongue, the way this is done varies widely. From what I've experienced, mangled grammar creates an involuntary mental block for those of us used to listening to / reading perfect English, but in itself is not hard to understand. While everyone has their personal brand of English grammar here, comprehension isn't obstructed (much), since that mental block doesn't occur to those used to random mangling of grammar.
In some cities (eg Bombay), an additional problem can be the inclusion of random Indic words in sentences. While most try to confine themselves to English while talking to Caucasians, I've witnessed a couple of hilarious incidents where someone unwittingly uses some Hindi in their sentences to a foreigner.
So the only course of action I see here is to just keep talking to others. Get rid of that mental block for bad grammar, and try to glean as many phrases as possible. It took me a couple of months to both get comfortable with spoken Hindi and Indian English; if you make a conscious effort, it might take much less time.
Problem #2 - words and phrases that I use, where no-one has the slightest clue what I'm on about! Sometimes it's fine, sometimes I have to have a few goes at rephrasing it to get a version that's understood, and sometimes it all seems fine but later I discovered they understood something completely different!
This is one of my personal pet peeves — there are a lot of idioms (et cetera) that I like to use, but nobody gets them. The way to fix this is simple in essence, though possibly hard in execution:
Treat Indian English as a completely separate language
I personally do this. My vocabulary in the two are different (not even subsets). It's at a stage that my mind won't come up with confusing sentences while I'm talking with Indians unless I explicitly ask it to. I still use a slightly higher level vocabulary than some are used to, but that's due to a change of my peer group which I haven't had a chance to fully adjust to1.
This ought to happen automatically as you start talking with others, but if you keep a conscious track of it, it happens much faster. You won't lose the ability to speak in your original British "language", either (It's like learning to ride a bike!)
1. Currently I'm in college with a larger variety of people (location wise) than in the past. Approximately half the country uses Hindi much more than English, including as a link language, which means that there are many people for whom English is a third or neglected second language. The roles for Hindi and English swap in certain cities and most of the South. So, depending on where you are, the average vocabulary level changes.