Этот цвет подходит к твоей новой рубашке.

Because цвет is a masculine singular noun. The demonstrative этот must agree with the noun in gender, number, and case.

In the nominative singular:

• masculine: этот
• feminine: эта
• neuter: это

So этот цвет is the correct form for this color.

Подходит is the 3rd person singular present-tense form of подходить.

It is singular because the subject is цвет, which is singular. So literally it is something like:

• цвет подходит = the color suits / matches / goes with

Russian present tense often covers meanings that English expresses with goes with, matches, or suits.

In this meaning, подходить к + dative means to go with, to match, or to suit something.

So:

• подходить к рубашке = to go with a shirt
• подходить к обуви = to match the shoes

This is just the normal pattern of the verb in this context.

Be careful: подходить can also be used without к in other meanings, for example:

• Мне это подходит = This suits me / works for me

So the grammar depends on the meaning.

Because the preposition к requires the dative case.

The dictionary form is рубашка, but after к it changes:

• nominative: рубашка
• dative: рубашке

So:

• к рубашке = to / for / with the shirt in this idiomatic sense of matching

Because adjectives and possessive words in Russian must agree with the noun they describe.

Here the noun is рубашке, which is:

• feminine
• singular
• dative

So the words describing it must also be feminine singular dative:

• твоя → твоей
• новая → новой
• рубашка → рубашке

That is why the whole phrase is:

• к твоей новой рубашке

Yes. Твоей comes from твой, which is the informal singular word for your.

So this sentence is addressed to one person in an informal way.

If you wanted to be formal or address more than one person, you would use вашей:

• Этот цвет подходит к вашей новой рубашке.

So:

• твой / твоя / твоё = your, informal singular
• ваш / ваша / ваше = your, formal or plural

Russian does not have articles like the or a/an.

Whether something is definite or indefinite is usually understood from:

• context
• word order
• demonstratives like этот (this)

In this sentence, этот already makes the noun specific, so этот цвет clearly means this color.

Yes. Russian word order is more flexible than English because the endings show the grammatical relationships.

The original sentence:

• Этот цвет подходит к твоей новой рубашке.

is a neutral, natural order.

You could also say:

• К твоей новой рубашке подходит этот цвет.

This puts more focus on your new shirt, as if answering What goes with your new shirt?

So the basic meaning stays the same, but the emphasis can shift.

Sometimes, but the meaning changes a bit.

• подходит к usually means matches / goes well with another item
• идёт often means looks good on someone or flatters someone

So here:

• Этот цвет подходит к твоей новой рубашке.
  = the color matches the shirt

But:

• Тебе идёт этот цвет.
  = this color looks good on you

English often uses suit for both ideas, but Russian tends to separate them more clearly.

The standard stress is:

• Э́тот цвет подходи́т к твое́й но́вой руба́шке.

A few useful notes:

• Э́тот
• подходи́т
• твое́й
• но́вой
• руба́шке

Stress matters in Russian, so it is worth learning words together with their stressed syllables.

Цвет is in the nominative singular because it is the subject of the sentence.

So the structure is:

• Этот цвет = subject
• подходит = verb
• к твоей новой рубашке = prepositional phrase with к + dative

In other words, the color is the thing doing the matching.