In my opinion, I think so has two main meanings in English, either it's about an opinion of the speaker, in which case you'd use the verb tycker in Swedish (det tycker jag), or a belief the speaker holds, which would be det tror jag, whereas I suppose so refers to an assumption the speaker makes, which would be jag antar det.
Is she a good singer? Yes, I think so -> Ja, det tycker jag (it is my opinion)
Will it rain tomorrow? Yes, I think so -> Ja, det tror jag (I believe it will).
Is that true? I suppose so -> Jag antar det (I don't really know, I'm just assuming here, but my assumption does not quite amount to a belief)
I think one could learn Swedish just by following your comments… I don't have enough lingots to thank you (and don't think you care about those really, they are useless unless there's some hidden option for duolingo active teachers to exchange them for money). What I'd like to have is a simple way to read your publications as a blog or a series of publications with one URL and a meaningful index. Whatever, you're my hero =)
Thanks a lot! I think periodically of how we should organize our explanations better (there's the sticky post under Swedish Discussions, but it only links to longer explanations) and try to write longer texts about some issues, but I guess it'll take some time until I/we get there. I hadn't thought about a blog but now you mention it, that isn't a bad idea either. I'll keep it in mind as a possibility for the future, but you'll probably be fluent in Swedish by the time I get my stuff organized :D
Det antar jag is totally ok too. The important thing is to have the verb in second place, but whether you want to have det or jag first is only a question of how you want to structure the utterance.
The general principle for this is called the information principle: you usually want to start out with known information and put new information later in the sentence. Or you can say that the thing you put first in the sentence is the "topic" – what the sentence is about, and the latter part is the "comment" – what you say about the topic.
It's often natural and neutral to start out with a personal pronoun, since 'you' are a given in any sentence 'you' speak. So in a way, Jag tror det or Jag antar det are very neutral sentences. You're saying something about your beliefs.
But in many contexts, you're referring back to something that has just been mentioned. This is the det in the sentence. So from that point of view, it's very natural to put det first. Now you're saying something about 'it' – that you believe, assume, etc. it.
So tl; dr; both are fine and it's just a very small nuance. :)