Hm...
puta -- whore
tangina -- short for "Putang ina". Putting an 'ng' after an adjective ending in a vowel is mandatory, followed with the noun that it describes. 'Ina' is mother. So basically this is "your mother is a whore", or as commonly said in English, "son of a bitch". Extra: if an adjective ends with a consonant, we add the word (not suffix) 'na' after the adjective. E.G. Malamig na balat (cold skin)
gago -> the closest English word for this is "fool", but the weight of the word here is definitely greater than the English word. It's actually considered a swear word.
tanga -> I think this is actually slang, but I've never heard "mangmang" used anymore now. This literally means "stupid". This word usually gets used more often as an expression rather than to describe someone though, so it's usually used alongside swearing, though it's not really a swear word in itself.
paksityet -- what? I think you may have heard "paksiyet" instead. "paksiyet" is a PH-bastardized version of "Fuck shit".
kantot -> I'm almost positive that this is slang for "fuck". It doesn't actually get used like the English equivalent. In normal conversations with the usual obscenity, this word is actually never used, and when it is, it's actually used for its actual meaning.