I like to read. - Мне нра́вится читать.
I will like to read. - Мне бу́ду нра́вится читать.
Here is a mistake.
[Мне будет] нравится читать.
There is no [Мне буду] in Russian.
Threre is only [Мне будет ...] or [Я буду ...].
I did like to read. - Мне был(а) нра́вится читать.
In contemporary Russian there is no usage of [быть] as tool-verb for the Past.
There was form which is not not used any more, but it is more not like tool-verb it is adverb.
For example:
[Мне бывало нравилось читать.]
If you say that in Russian nowdays they will understand you, but usually this form is pretty book-ish.
[I did like] instead of [I liked] is the same as [I really liked]
[I did like] = [I really liked]
Therefore in Russian it is
[I did like to read.] = [Мне реально нравилось читать.]
I want to repair ... - Я хочу́ почини́ть...
Possible options:
[Я хочу починить.]
[Я буду хотеть починить.]
[Я хотел починить.]
Or based on self-form of the verb.
[Мне хочется починить.]
[Мне будет хотеться починить.]
[Мне хотелось починить.]
I worked on repairing the transmission.
Я порабо́тал чини́ть переда́чу.
You can not use [поработал + verb].
В по́лдень я рабо́тал: чини́л переда́чу моего́ пика́па.
If your task was to express idea that yesterday you've started to repair, but till now you have not repaired it yet and you need more time to finish your job, then [В по́лдень я рабо́тал: чини́л переда́чу моего́ пика́па] does not say about that anything.
[:] - is used to explain, to give more detais.
You can say.
[В по́лдень я рабо́тал.]
And I can ask: what exactly did you do?
You can answer:
[Я чини́л переда́чу моего́ пика́па.]
So, you explained me the deep meaning of what your working was about.
In order to avoid my "investigation" question, you can beforehand to give detais by yourself like this:
[В по́лдень я рабо́тал. Я чини́л переда́чу моего́ пика́па.]
That is the basic and the full form for this case.
There are 2 ways, how you can "pack" these 2 sentances in Russian language.
#1. Full package.
[В по́лдень я рабо́тал, а именно: я чини́л переда́чу моего́ пика́па.]
#2. Short version of the full package.
[В по́лдень я рабо́тал: чини́л переда́чу моего́ пика́па.]
In this case #1 is 100% equal to #2 and they both equal just to pair of standalone sentances.
The first says some general information: you we working.
The second one gives more details on what exaclty you were doing - you we doing the repair.
There is no need in Russian language to mix [работал] + [verb].
It just does not make sence.
If your feelings drive you to form stuff like this:
[Я работал чинить передачу.]
Then no, that pattern does not used in Russian, not because it is technically impossible, but just because there is no sence or need in using such a pattern.
In Russian language it is 100% enough to say.
[Я чинил передачу.]
In this case everybody 100% understand, that you were working, just because it is phisically impossible to repair your car and not being under working.
Therefore, if you want to check if you can mix two verbs or you can't just think if the first verb is already included in the second by meaning of action in second verb.
Let's take any random verbs, just from the top of the head.
[плавать] = swim
[дышать] = breathe
Я плаваю дышать. = I swim breathe.
Я буду плавать дышать. = I will swim breathe.
Я плавал дышать. = I swam breathe.
For me this direct mixing sounds crazy both in English and in Russian, but as you can see, technically we can build this stuff.
Let's make it more real.
Я плаваю, дыша. = I swim breathing.
Я буду плавать, дыша. = I will swim breathing.
Я плавал, дыша. = I swam breathing.
In you case
[Я работал, чиня передачу.]
It is possible and correct form in Russian language, but it is too complicated and real simple people try to avoid such complex patters.
- Honey, where were you?
- [Работал. Чинил передачу.]
Or as your teacher recommended.
- Honey, where were you?
- [Работал: чинил передачу.]
(Because I finished the work [say, yesterday], and I did not accomplish my task)
If you want to express idea, that you have started reparing, but you still have a lot of to do and you even do not have idea when exactly you gonna finish.
В полдень я начал чинить передачу моего пикапа.
At noon I started (began) to repair {my car}.
Possible options.
Я начинаю чинить.
Я буду начинать чинить.
Я начал чинить.
Bonus track
[В полдень] = [At noon] = [12:00 a.m.]
[Вчера в полдень] = [Yesterday at noon] = [Yesterday at 12:00 a.m.]
[Днем] = [In the afternoon]
[Вчера днем] = [Yesterday in the afternoon]
In Russian and I guess in English too they do not say
[I repaired my car at noon.]
They might say
[I began to repair (started repairing) my car at noon.]
At noon is just a tiny moment in time which is 12 hours 00 minutes 00 seconds.
[I began to repair my car at noon.] = ok
[I began to repair my car in the afternoon.] = ok
[I repaired my car in the afternoon.] = ok
[I repaired my car at noon.] = weird.
Well, it is weird for your case, but there can be situation where this phrase can be ok.
If someone was killed at 12:00 and police officer during investigation might ask you:
Where were you at noon (at 12:00)?
You may answer:
[I repaired my car at noon.] = [I repaired my car at 12:00.],
and my coworkers can confirm that, so I have an alibi and therefore I am not the murder.