I guess that was a much too short answer. I meant this: på is the standard preposition for days of the week, just like i is the one for months. på måndag but i januari. So på is the default preposition for saying something took place on Tuesday.
Of course other prepositions can be used for other meanings: före tisdag for 'before Tuesday' or under tisdagen for 'during Tuesday' or efter tisdag for 'after Tuesday' and so on.
It is also of course possible to mention the day without a preposition as long as you're not saying that something took place on that day: Idag är det tisdag 'It's Tuesday today'.
But when we want to say that something takes place on a day, we want a preposition. In English, you can hear people say I'll do it Monday, without a preposition, but that doesn't work in Swedish.
However, förrän is a conjunction [subjunction in newer terminology]. It establishes the relationship between phrases or groups of words, but it does not anchor the event sufficiently in time for our taste. This seems to be the case for all conjunctions in Swedish, although one på can be shared across an och or eller: Han kommer på måndag eller tisdag – you don't need to add an extra på for Tuesday since it can share the one Monday has. (it wouldn't be wrong to add it though).
innan is the tricky case, it could be either a preposition or a conjunction. However when it's followed by på tisdag, it functions as a preposition, so we don't need (or want) one more to anchor the action in time.
Just to make things just a little more complicated, if you add a modifier such as nästa, 'next', the preposition will drop again. Jag kommer på måndag but Jag kommer nästa måndag (you can't add på to the latter one). This is pretty parallel to English though so it shouldn't be too strange.
tl;dr förrän is a conjunction so you need a preposition, the others are prepositions so you don't need one more.
After reviewing comments, I'll come from far right field on this. I'm from California, my mother from Manhattan. I learned from her.
I don't think I use "to swim" to alert people to my intent to swim. Only to the type of swimming: I swim well; I swim the backstroke but not the butterfly, etc.
We combine the verb "to go" and the noun "swimming" or "a swim" as follows:
I am about to go for a swim. I am about to go swimming. I will go swimming whenever. I went swimming earlier.
Any thoughts? Regional American usage?
It isn't currently an accepted translation, but in principle you could argue it should be, like this: This English sentence is ambiguous. Either it means what we first think it means: 'Tuesday is the earliest point in time I will go swimming'. But it could also be interpreted like this: 'I am not in the process of swimming from today all the time until Tuesday'. The latter interpretation would use tills. However this interpretation of the sentence, without any context to back it up, is far-fetched, and we usually don't accept such translations, since accepting them could be misleading – it could lead people to think that you could use tills for the most natural interpretation of this sentence, but you can't.