По-моему, цена высокая.

По-моему means in my opinion, to my mind, or I think.

So:

• По-моему, цена высокая.
• In my opinion, the price is high.

It is a very common conversational way to give your opinion.

Because this expression is conventionally written that way in Russian.

• по-моему
• по-твоему = in your opinion
• по-нашему = in our opinion / our way

This is a fixed spelling pattern for this kind of adverbial expression.

Because по-моему is being used as a parenthetical opinion phrase, like in my opinion in English.

So the comma shows that it is separate from the main statement:

• По-моему, цена высокая.
• In my opinion, the price is high.

You may also see it later in the sentence:

• Цена, по-моему, высокая.

That means the same thing.

In Russian, the verb to be is usually omitted in the present tense.

So:

• цена высокая literally looks like price high but means the price is high

This is completely normal in Russian.

If you wanted past or future, then a form of быть would appear:

• Цена была высокой. = The price was high.
• Цена будет высокой. = The price will be high.

Because цена is a feminine noun, and the adjective has to agree with it.

• цена = feminine, singular
• so the adjective must also be feminine, singular: высокая

Compare:

• высокий = masculine
• высокая = feminine
• высокое = neuter
• высокие = plural

So:

• цена высокая
• дом высокий = the house is tall/high
• здание высокое = the building is tall
• цены высокие = the prices are high

Because it is the subject of the sentence.

In цена высокая, the structure is basically:

• цена = the price
• высокая = high

Since цена is what is being described, it stays in the nominative.

Not exactly.

Высокая цена means a high price and is the most natural match for English high price.

Дорогой / дорогая usually means expensive, and it more naturally describes the thing being sold, not the price itself:

• Это дорогая машина. = This is an expensive car.

You may hear дорогая цена, but высокая цена is generally the more standard and natural way to say a high price.

So in this sentence, цена высокая is very natural.

Yes. Russian word order is flexible.

All of these are possible:

• По-моему, цена высокая.
• Цена, по-моему, высокая.
• Цена высокая, по-моему.

They all mean roughly In my opinion, the price is high, but the emphasis shifts a little.

The version you were given is a very normal, neutral way to say it.

Моему is a form of мой (my).

In this expression, it is in the dative singular neuter form, but for learners it is best to treat по-моему as a fixed expression meaning in my opinion.

Useful parallels:

• по-моему = in my opinion
• по-твоему = in your opinion
• по-вашему = in your opinion / in your way
• по-нашему = in our opinion / our way

So although the grammar behind it is real, the easiest approach is to learn по-моему as a set phrase.

Yes, in some contexts it can mean in my way or the way I do, depending on the sentence.

But in По-моему, цена высокая, it clearly means in my opinion.

For example:

• Сделай по-моему. = Do it my way.
• По-моему, это ошибка. = In my opinion, this is a mistake.

So context tells you which meaning is intended.

You can say:

• По-моему, цена низкая.

Here низкая means low, and it is feminine singular to agree with цена.

So the pattern is the same:

• По-моему, цена высокая. = In my opinion, the price is high.
• По-моему, цена низкая. = In my opinion, the price is low.