'Tror' is also something you think but aren't sure about e.g. 'I think the film is good (but I haven't seen it yet so can't be sure)' while 'synes' expresses your opinion e.g 'I think the film is good (because I have seen it)'. So 'tror' expresses uncertainty while 'synes' expresses opinion.
tror is about belief, trust and faith. Jeg tror på hende. (I believe in her/I have faith in her.) Jeg tror på Gud. (I believe in God.) Jeg tror det regner. (I think it's raining (right now).)
tænker is about intending and pondering. Jeg tænker vandre til Sverige. (I will walk to Sweden/I'm intending to walk to Sweden.) Jeg tænker på livet. (I think/ponder about life.) Jeg tænker på dig. (I'm thinking about you.)
synes is about opinion and impression. Jeg synes det er godt. (I think it's nice/My opinion is that it's nice.) Jeg synes han er en smug fyr. (I think he's a great guy.)
It is tricky in the beginning, but you have to think of it as a whole new concept, rather than translating from directly English.
erikblomqvist The Danish for I'm considering walking to Sweden (which would be hard, because I don't think you can walk over "Broen", and usually during an isvinter, they clear en skibsrende i Øresundet--a ship's path!) is Jeg tænker på at vandre til Sverige.
Or you could write Jeg tænker på at udvandre til Sverige. (I'm thinking about emigrating to Sweden)
Or Jeg tænker på at gå på vandring i Sverige." I'm thinking of going for a hike in Sweden.
And in none of these could you use tror, mener, or synes, which are all about opinion.
On the other hand you could say Jeg tænker på hende hele tiden! (I think about her all the time.)