In non-formal conversation this is fine, except it needs more articles:
a new life, because you are not starting new-life-in-general but a particular (albeit unspecified) new life, which presumably you are going to describe.
the next week, because bare next week always means 'next' with respect to the time when you are speaking.
It's barely acceptable in formal use, too; but as a matter of courtesy to your readers it is desirable to pin down your time references a little more precisely. Otherwise readers will see was taking and assume that it refers to that day, and then have to revise that interpretation when they come to the next week. Write this instead:
That day I started a new life. I would be taking Spanish classes the next week, and &c
Effective communication is driven by the principal of minimal effort; but that means very different things in written and spoken English, because the two media present radically different constraints:
Written English is governed by The Adamantine Law: Whatever can be misunderstood will be.
When you write you have no opportunity to correct your readers' misunderstandings. Consequently you must take care to express your thoughts as clearly and unambiguously as possible, so no misunderstandings arise. It is your reader's effort which must be minimized, not your own.
Spoken English, in contrast, is governed by The Tolerance Maxim:
Whatever should be understood may be omitted.
When you speak you are engaged in a conversation. You and your interlocutors share extensive knowledge of the situation which need not be specified, you may speak elliptically and allusively in confidence that your interlocutors will fill in the holes and tacitly ignore or adjust any syntactical or grammatical blunders you commit. If significant misunderstandings arise, they may be corrected. All parties share the effort.
Using the present tense for future events indicates certainty, consistency, and familiarity. In other words, use this to talk about events which will happen, which happen on a regular basis (or are predictable in some way), and about which you have some personal knowledge.
The holidays start next week.
I know this happens every year, last year I saw it happen in the same way, and I am sure it will happen the same way this year.
The train leaves in five minutes
I know the train leaves at a specific time, I know that trains usually leave on time, and I have seen the schedule to confirm when the train will leave.
The gala event starts tomorrow
I have personal knowledge that it is scheduled for tomorrow, I know that events like this usually start on schedule, and I feel comfortable asserting that it will definitely happen.
If any of these three does not exist, there is uncertainty, and the future tense will be more appropriate.
We get paid on Friday
I know this happens with some consistency and regularity, as I've already received at least one paycheck on Friday.
We will get paid on Friday
Oops, now there is some doubt. I hope this will happen, but I'm not sure it'll happen.
Here's another example: given no other context, consider these two sentences:
He will make the winning shot of the game in twenty seconds.
He makes the winning shot of the game in twenty seconds.
The first indicates a future condition, but the context is unclear. Do I really know this will happen? Probably not. More likely I'm expressing an opinion, or a personal hope.
But the second sentence, that expresses an odd certainty. The context is either that we are watching a recording of the game which I have seen before, or that I'm psychic, and I've already seen this happen with my mental powers.
Best Answer
The first two are wrong.
The second pair is awkward.
The following is largely idiomatic, but it just might point you in the right direction:
That said, informal speech relies heavily on contractions.