Σήμερα θα φτιάξω μελιτζάνα και κολοκύθι στο ταψί με λίγο ελαιόλαδο.

θα is the particle that marks the future in Modern Greek.

So:

• φτιάχνω = I make / I am making
• θα φτιάξω = I will make / I’ll prepare

Unlike English, Greek does not use a separate verb like will. Instead, it uses θα plus the appropriate verb form.

Because θα φτιάξω presents the action as one complete event: Today I’ll make/cook ...

Very roughly:

• θα φτιάξω = I will make / I’ll prepare
  
  • a single, complete action
• θα φτιάχνω = I will be making / I will make regularly
  • ongoing, repeated, or habitual action

In this sentence, the speaker is talking about making one dish today, so θα φτιάξω is the natural choice.

Yes. In everyday Greek, φτιάχνω is very common and very broad. It can mean:

• make
• prepare
• fix
• and in food contexts, often cook

So θα φτιάξω μελιτζάνα... naturally means I’ll make / cook eggplant...

You could also use μαγειρεύω for cook, but φτιάχνω is extremely common in casual speech when talking about food.

Greek often uses the singular when naming ingredients or types of food in a general way.

So here:

• μελιτζάνα
• κολοκύθι

sound like eggplant and zucchini/squash as ingredients, not necessarily one exact eggplant and one exact zucchini.

English can do something similar: Today I’m making eggplant and zucchini...

If you wanted to focus more on actual individual vegetables, the plural could also appear in other contexts:

• μελιτζάνες και κολοκύθια

But the singular in your sentence is very natural for a recipe-style idea.

Because they are being used in a general, ingredient-like way.

Greek uses articles more often than English, but it can still leave them out in certain contexts, especially with:

• foods
• ingredients
• materials
• recipe-style wording

So:

• θα φτιάξω μελιτζάνα και κολοκύθι = a general statement about what is being prepared

If you added articles, it would sound more specific:

• θα φτιάξω τη μελιτζάνα και το κολοκύθι
  = I’ll make the eggplant and the zucchini
  meaning some specific ones already known in the conversation

στο is a contraction of:

• σε + το = στο

So στο ταψί literally means:

• in the baking tray
• in the pan
• sometimes more naturally in English, tray-baked or in a roasting dish

ταψί is a baking tray / roasting pan, and στο ταψί is a very common cooking expression in Greek.

It does not simply mean in the oven.
For in the oven, Greek would usually say:

• στον φούρνο

So:

• στο ταψί = in the tray/pan
• στον φούρνο = in the oven

Because λίγο works as a quantifier meaning a little or some, and ελαιόλαδο is an uncountable noun.

So:

• με λίγο ελαιόλαδο = with a little olive oil

This is the normal pattern.

If you added an article, the meaning would become much more specific:

• με το λίγο ελαιόλαδο
  = with the little olive oil / with that small amount of olive oil

That would only make sense in a special context where a specific amount of olive oil had already been mentioned.

They are in the accusative, because they are either:

• direct objects of the verb, or
• objects after prepositions

More specifically:

• μελιτζάνα and κολοκύθι are the direct objects of θα φτιάξω
• ταψί is after σε → στο
• ελαιόλαδο is after με

A useful thing to notice is that in this sentence, several forms look the same as the dictionary form, especially because:

• some nouns do not visibly change much
• articles are omitted in some places

So the case is there grammatically, even if the noun form itself does not change very obviously.

They are:

• η μελιτζάνα = eggplant/aubergine, feminine
• το κολοκύθι = zucchini/squash, neuter
• το ταψί = baking tray/pan, neuter
• το ελαιόλαδο = olive oil, neuter

This matters because adjectives, articles, and pronouns have to agree with the noun’s gender, number, and case.

Greek word order is fairly flexible.

This sentence begins with Σήμερα because the speaker wants to set the time frame first:

• Σήμερα θα φτιάξω... = Today I’ll make...

You could also say:

• Θα φτιάξω σήμερα μελιτζάνα και κολοκύθι...

That is also grammatical, but the emphasis shifts slightly.

Putting Σήμερα first is very natural when today is the starting point of the message.

Yes. Here και just links the two ingredients:

• μελιτζάνα και κολοκύθι = eggplant and zucchini

It works very much like English and in this sentence.