Iulius calidior est quam Iunius, et mensibus Iulio atque Augusto multi ad mare proficiscuntur.

Here they are the months July and June. Latin month names can look exactly like personal names, so context tells you which meaning is intended.

In this sentence, the comparison with calidior makes it clear that the words refer to months:

• Iulius = July
• Iunius = June

Calidior is the comparative form of calidus, meaning warmer rather than just warm.

So:

• calidus = warm
• calidior = warmer

Latin usually forms comparatives with:

• -ior for masculine/feminine
• -ius for neuter

Because Iulius is masculine singular, the adjective is calidior.

Because Iulius is a masculine noun. The comparative adjective has to agree with the noun it describes.

So:

• masculine/feminine nominative singular: calidior
• neuter nominative singular: calidius

Since Iulius is masculine, Latin uses calidior.

Quam means than after a comparative adjective or adverb.

So:

• calidior ... quam Iunius = warmer than June

A useful thing to notice is that the word after quam is often in the same case as the thing being compared with it. Here:

• Iulius is nominative
• Iunius is also nominative

That is why you get quam Iunius, not some other case.

Both are nominative singular.

• Iulius is the subject of est
• Iunius is also nominative because it is being compared with Iulius after quam

So the structure is basically:

• July is warmer than June

They are all in the ablative.

More exactly:

• mensibus = ablative plural
• Iulio = ablative singular
• Augusto = ablative singular

Latin often uses the ablative of time when to mean in a certain time. So:

• mensibus Iulio atque Augusto = in the months of July and August

Iulio and Augusto are in apposition to mensibus, so they match the ablative idea.

Because Latin often does not use a preposition for time expressions like this. Instead, it uses the ablative of time when.

So Latin can simply say:

• Iulio = in July
• Augusto = in August
• mensibus Iulio atque Augusto = in the months of July and August

This is very normal Latin usage.

Latin could often just say Iulio atque Augusto by itself, and that would still mean in July and August.

Adding mensibus makes the time expression more explicit:

• Iulio atque Augusto = in July and August
• mensibus Iulio atque Augusto = in the months of July and August

So mensibus is not strictly necessary, but it is perfectly natural and helps spell out the idea.

Both et and atque mean and.

In this sentence:

• et joins the two main clauses
• atque joins Iulio and Augusto

So the structure is:

• first clause: Iulius calidior est quam Iunius
• second clause: mensibus Iulio atque Augusto multi ad mare proficiscuntur

Using atque here is completely normal. It often feels a little more closely linking than plain et, but in many sentences the difference is slight.

Because multi is an adjective being used substantively, meaning the noun is understood rather than stated.

So:

• multi literally means many
• but in context it means many people or many persons

Grammatically, multi is:

• nominative masculine plural
• the subject of proficiscuntur

Latin does this very often:

• multi = many people
• pauci = few people
• boni = good men / good people, depending on context

Because proficiscor is a deponent verb.

Deponent verbs:

• have passive-looking forms
• but active meanings

So:

• proficiscor = I set out / depart
• proficiscuntur = they set out / depart

Here proficiscuntur is:

• present tense
• indicative mood
• third person plural

So multi ad mare proficiscuntur means many people set out for the sea or many people go to the seaside.

Ad takes the accusative and expresses motion toward a place.

So:

• ad mare = to the sea / to the seaside

Mare is accusative singular here. It looks the same as the nominative because mare is a neuter third-declension noun, and neuter nominative and accusative singular often have the same form.

Also, ad mare is better than in mare for this meaning:

• ad mare = toward the sea, to the coast/seaside
• in mare = into the sea

All three can work, depending on how literal or natural you want to be.

• set out is close to the core meaning of proficiscor
• depart is also accurate
• go may sound more natural in smooth English

With ad mare, idiomatic English often uses:

• many people go to the seaside
• many people set out for the sea
• many people leave for the coast

So the exact English wording can vary, even though the Latin grammar stays the same.

Yes. In classical Latin, the letter J was not normally used the way it is in English. Latin often wrote I where English-based spelling might use J.

So:

• Iulius = Julius
• Iunius = Junius / the month June
• Iulio = in July
• Augusto = in August

So the spelling may look unusual to an English speaker, but it is standard classical Latin spelling.

It may feel unusual compared with English, but it is normal Latin.

A very literal order would be something like:

• July warmer is than June, and in the months July and August many to the sea set out

Latin word order is more flexible than English word order. Here the sentence is arranged quite naturally:

• first, the comparison between July and June
• then, the time phrase mensibus Iulio atque Augusto
• then the subject multi
• then the destination and verb: ad mare proficiscuntur

So the word order is Latin, but not strange Latin.