Is a lot of informal?

Mildly, and it is the default in speech at every level. In formal writing many is preferable for countables, and much is fine in negatives and questions. For a formal positive statement about an uncountable noun, a great deal of or considerable is the usual replacement.

What is the difference between much and a lot?

Only register and where they can sit. They mean the same, but much is restricted to negatives, questions and combinations with too, so, as and how, while a lot of goes anywhere. There is no meaning difference to learn, just a distribution one.

Do some nouns work both ways?

Yes, with different meanings. Hair is uncountable as a mass. She has long hair, and countable as individual strands: there are two hairs in my soup. Time, coffee, paper and experience all behave this way, so the noun alone does not settle it; the meaning does.

Which one to useA1

Much vs many

The short answer

Many goes with things you can count: many books, many people. Much goes with things you cannot: much time, much water. In positive statements both sound formal, and English speakers normally say a lot of instead: I have a lot of work, not I have much work.

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At a glance

Much or Many?

Compare	MuchUncountable. Cannot be pluralised.	ManyCountable. Has a plural.
Goes with	Singular uncountable nouns	Plural countable nouns
The test	Cannot say “two of them”	Can say “two of them”
Examples	much time, much money, much advice	many hours, many coins, many tips
In questions	How much does it cost?	How many are there?
In positive statements	Formal: use a lot of instead	Acceptable, but a lot of is commoner
With too / so / as	Natural: too much, so much	Natural: too many, so many

How to choose

Ask these, in order

1
Can you put a number in front of the noun?
Many. Three chairs works, so many chairs works.
How many chairs do we need?
Chairs can be counted one by one.
2
Is the noun a mass, something measured rather than counted?
Much. Water, time, money, information, advice, furniture.
We don’t have much time.
Time is measured, not counted.
3
Is the sentence positive, with no too, so or as?
Use a lot of. Much and many both sound stiff in plain positive statements.
There’s a lot of traffic this morning.
“There is much traffic” is grammatical and nobody says it.
4
Is the noun one English counts differently from your language?
Check it. Information, advice, news, furniture, luggage and money are uncountable in English and countable in many other languages.
She gave me a lot of useful advice.
Not “advices”, advice has no plural in English.

Side by side

Each one, in full

Much

When to use much

Uncountable. Cannot be pluralised.

Much goes with uncountable nouns: substances, abstractions and anything measured rather than counted. The reliable test is whether the noun has a plural: if you cannot say two of them, it takes much.

In modern English much is largely confined to negatives, questions and the fixed combinations too much, so much, as much and how much. In a plain positive statement it sounds noticeably formal or old-fashioned, which is why a lot of has taken over.

I didn’t get much sleep.

Negative, much is natural here.

How much did it cost?

A question about an amount.

There’s too much salt in this.

With too, much is the only option.

NBA lot of cannot follow too, so the fixed combinations keep much alive.

Many

When to use many

Countable. Has a plural.

Many goes with plural countable nouns and behaves more freely than much: it is acceptable in positive statements, though a lot of is still more common in speech.

Watch the nouns English counts differently from most other languages. Information, advice, news, furniture, luggage, equipment, work, research and money are all uncountable in English, so they take much even where their equivalents elsewhere are plural.

How many people are coming?

People is countable, despite having no -s.

Many of my colleagues work remotely.

Many of + a specific group.

Not many shops open on Sundays here.

Negative, and perfectly natural.

Mistakes to avoid

What learners actually get wrong

I have much work today.
I have a lot of work today.
Much in a plain positive statement sounds formal to the point of odd. It is not ungrammatical, which is why textbooks let it pass, but no native speaker would say it in conversation.
She gave me many advices.
She gave me a lot of advice.
Advice is uncountable in English and has no plural, so it takes much rather than many. The same applies to information, news, furniture and luggage, a list worth memorising because your own language probably counts them.
How much people were there?
How many people were there?
People is a countable plural even though it does not end in -s. Its singular is person, and the absence of the -s is what misleads learners into treating it as a mass noun.
There were too many traffic.
There was too much traffic.
Traffic is uncountable, so it takes much and a singular verb. Both halves of the sentence have to change together, which is why this error tends to come in pairs.

Now use it

Countable or uncountable?

Sort the nouns, and stop saying informations and advices. Reading a rule and being able to apply it are different skills, and only one of them shows up when you speak.

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Questions

Asked and answered

Is a lot of informal?: Mildly, and it is the default in speech at every level. In formal writing many is preferable for countables, and much is fine in negatives and questions. For a formal positive statement about an uncountable noun, a great deal of or considerable is the usual replacement.
What is the difference between much and a lot?: Only register and where they can sit. They mean the same, but much is restricted to negatives, questions and combinations with too, so, as and how, while a lot of goes anywhere. There is no meaning difference to learn, just a distribution one.
Do some nouns work both ways?: Yes, with different meanings. Hair is uncountable as a mass. She has long hair, and countable as individual strands: there are two hairs in my soup. Time, coffee, paper and experience all behave this way, so the noun alone does not settle it; the meaning does.