Vores træner siger, at styrketræning og udstrækning begge er vigtige, hvis man vil undgå ømhed.

After possessives like min, din, vores, Danish uses the noun in the indefinite form:

• vores træner = our coach
• træneren = the coach

So vores træneren is not correct. The possessive already makes the noun definite in meaning.

Danish writes compound nouns as one word much more often than English does. Styrketræning is made from styrke + træning, literally strength-training.

This is very normal in Danish:

• styrketræning
• fodboldklub
• sommerferie

So if English uses two words, Danish will often use one compound noun instead.

Both at and hvis introduce subordinate clauses. This sentence uses startkomma, a punctuation style where you put a comma before a subordinate clause:

• siger, at ...
• vigtige, hvis ...

In modern Danish, many writers omit these commas. So a version without them can also be correct, depending on the comma system being used.

Here at is the conjunction that. It introduces what the coach says:

• Vores træner siger, at ... = Our coach says that ...

English often drops that, but Danish very often keeps at after verbs like sige, tro, mene, and vide.

Because Danish uses different word-order rules in main clauses and subordinate clauses.

• In a main clause, the finite verb normally comes in second position.

  • Vores træner siger ...

• In a subordinate clause introduced by at or hvis, that rule does not work the same way. The subject comes before the finite verb.

  • at styrketræning og udstrækning begge er vigtige
  • hvis man vil undgå ømhed

This main-clause vs. subordinate-clause difference is a very important part of Danish syntax.

Begge means both. It tells you that the statement applies to the two things mentioned: styrketræning and udstrækning.

Yes, you can also say:

• Både styrketræning og udstrækning er vigtige.

That version is very common and often sounds a little more natural in everyday speech. The given sentence is using begge instead.

Because the subject is plural: two things are being talked about, styrketræning and udstrækning. Predicate adjectives in Danish agree in number:

• Det er vigtigt. = It is important.
• De er vigtige. = They are important.

So here you need the plural form:

• styrketræning og udstrækning ... er vigtige

Yes. In Danish, man is an impersonal pronoun meaning one, you, or people in general.

So:

• hvis man vil undgå ømhed

means something like:

• if you want to avoid soreness
• if one wants to avoid soreness

It is not specifically masculine, even though it looks like English man.

Here it means want to or intend to, not simple future.

So:

• hvis man vil undgå ømhed

is best understood as:

• if you want to avoid soreness

Danish vil can express future, intention, or willingness, so context decides the best translation.

After a modal verb like vil, Danish uses the bare infinitive:

• vil undgå
• kan træne
• skal arbejde
• må gå

So:

• undgå is correct
• undgår would be a finite present-tense verb, which does not fit after vil
• at undgå is also wrong after a modal verb

This is similar to English will avoid, can train, must go.

Because these nouns are being used in a general or abstract way.

• styrketræning = strength training in general
• udstrækning = stretching in general
• ømhed = soreness in general

Danish often leaves out the article when talking about activities, abstract ideas, or things in general:

• Kaffe er dyrt.
• Motion er sundt.
• Ømhed kan være normalt.

If you meant a specific instance, you would be more likely to use an article or another determiner.