Translation:I think that Mr. Tanaka's mother is a teacher.
~様 is on a much higher politeness level than ~さん. It doesn't really have a direct English equivalent or at least I don't know how you would translate it without sounding cheesy or ingratiating. You wouldn't use ~様 in every day speech to regularly address colleagues etc. ~様 is not to be used to address old "Joe Blow" - ie. not for just anybody.
There's no equivalent for ~ちゃん/くん you've got that right. Maybe you could convey it in English by changing the person's name to show the speaker's familiarity with the person they're addressing eg. サムくん could be translated as Sammy in English, but then if you're translating for school or for work there's the issue of how much lee-way a teacher or supervisor would allow you in terms of translation.
Actually, it's "だ と", two separate words/particals. The だ here is the plain form of です. If the sentence lacked "I think", you would write "田中さんのお母さんは先生です". But because we add "I think", we need to say the sentence in plain form.
と思う is the particle と and the verb. You'll see this many times in Japanese! Duolingo also teaches と言う, and the grammar for と思う and と言う are similar.
I left off the Mr. entirely since that's how I translate x-san in my head, always. After all, yeah, you can use it with personal names, and it doesn't have a gender, etc. It gets thrown in always so you have to deduce from context when it means something in English and when you can leave it out. It also depends on the level of politeness. When I see "I think" at the end of a sentence, it often implies a higher level of humbleness so I probably should have left the Mr. (or Ms.) in.
You're casting very broad assumptions about the age of both Mr/Ms Tanaka's mother and Ms/Mr Tanaka. How old do you think you have to be to be a parent? Some kids have babies in their teens - some as young as 12! They definitely wouldn't be retired. I'm in my 40s with 3 children - not anywhere near retirement.
To AnaLydiate. You seem a bit fixed on using a title. It's safe to assume the speaker knows Tanaka, and the speaker is not addressing Tanaka. We don't know how old they are. When I was young, private school kids always addressed each other by surname only, no title. Titles were only used when directly addressing adults. (At the time this was a bit different from most people). These days that seems to have spread to the general population. So I have to disagree, and say it is now general practice to use surname only in English, for direct and indirect addressing. But of course first names and titles are also still optional, depending on the context. And re margaret and whether Mrs Tanaka's mum would be retired. Without context your reply is quite correct, who knows how old either are? But in the modern world, 12 year-olds having children is definitely frowned upon. Also the average age of first-time marriages, and motherhood is getting older, as it is normal now for women to have a career. (Very different from when I was young). Therefore I think margaret's assumption is completely understandable, though still an assumption. If we are using "Mrs" Tanaka, then the question is not, does her mum have children, but does she have married children. And is poor old mum still slaving away 9 to 5, or has she been able to retire? (Also an assumption).
"say it is now general practice to use surname only in English" - really? In what country?? Maybe in the military, but I honestly can't remember a single time anybody addressed me by my surname alone since high school. (Also I think you were replying to the wrong comment here...)