Predlažem piknik u parku.

Because Serbian usually does not need a subject pronoun when the verb ending already shows who is doing the action.
Predlažem already means I suggest.

You can add ja if you want emphasis or contrast:

• Ja predlažem piknik u parku. = I am suggesting a picnic in the park.
• Ja predlažem, a ti odluči. = I suggest it, and you decide.

So in a neutral sentence, leaving out ja is completely normal.

Predlažem is the 1st person singular present tense form of the verb predlagati = to suggest / to propose.

The ending -em tells you it means I:

• predlažem = I suggest
• predlažeš = you suggest
• predlaže = he/she/it suggests

A useful extra point: the perfective partner is predložiti, but the form in your sentence is the imperfective predlažem, which is the normal present-tense form for I suggest.

Serbian has no articles. So nouns often appear without anything corresponding to English a/an or the.

That means:

• piknik can mean a picnic or the picnic
• u parku can mean in a park or in the park

The exact meaning depends on context. If Serbian speakers want to be more specific, they can use other words, such as:

• jedan = one / a certain
• taj = that / that specific

But very often nothing is added.

Yes, piknik is the direct object, so it is in the accusative case. But it does not visibly change here because piknik is a masculine inanimate noun.

For many masculine inanimate nouns in Serbian:

• nominative singular = accusative singular

So:

• piknik (subject form)
• piknik (object form)

They look the same.

Compare that with a masculine animate noun, where the accusative usually changes:

• student → vidim studenta = I see the student

Because the preposition u uses different cases depending on the meaning:

• u + locative = location, in / at
• u + accusative = movement, into / to

In your sentence, the picnic is located in the park, so Serbian uses the locative:

• u parku = in the park

Compare:

• Piknik je u parku. = The picnic is in the park.
• Idemo u park. = We are going to the park.

So u parku is correct because this sentence describes a place, not movement.

Parku is the locative singular form of park.

This is very common with masculine nouns ending in a consonant:

• park → u parku
• grad → u gradu
• hotel → u hotelu

The locative is often used after prepositions such as:

• u = in
• na = on / at
• o = about

So u parku is a very typical Serbian pattern.

No, Serbian word order is fairly flexible.
Predlažem piknik u parku is a neutral, natural order.

But you can move things around for emphasis:

• U parku predlažem piknik. = emphasizes in the park
• Piknik u parku predlažem. = more marked or contrastive

Even though word order can change, the original sentence is the most straightforward and natural choice for a learner.

Yes, it is natural and correct. But in real conversation, Serbian speakers may also use longer or slightly different patterns, depending on what exactly they mean.

For example:

• Predlažem da idemo na piknik u parku. = I suggest that we go for a picnic in the park.
• Predlažem piknik u parku za vikend. = I suggest a picnic in the park for the weekend.
• Hajde na piknik u parku. = Let's go for a picnic in the park.

So your sentence is perfectly good, but Serbian also often uses predlažem da... with a full clause.

The letter ž sounds like the s in measure or the g in genre.

A rough English-friendly pronunciation of the whole sentence is:

pre-DLA-zhem PIK-neek oo PAR-koo

That is only approximate, but it will help you get started.