V práci mám nového kolegu, který plánuje velký projekt.

Because nového kolegu is in the accusative case, used for the direct object of the verb mám (I have).

• Subject (nominative): kolega je nový – The colleague is new.
• Object (accusative): mám nového kolegu – I have a new colleague.

For masculine animate nouns like kolega, both the noun and the adjective change in the accusative:

• nový kolega (nominative) → nového kolegu (accusative).

Kolega is a bit tricky: it ends in -a, but it is masculine animate.
Its declension is special:

• Nominative: kolega – a colleague (subject)
• Accusative: kolegu – a colleague (object)

So in mám nového kolegu, kolegu is the accusative form required after mám. The ending -a in nominative does not always mean a word is feminine in Czech.

The preposition v meaning in / at (a place) normally takes the locative case.
The noun práce (work) in the locative singular is práci. So:

• práce (nominative) – work (as a thing)
• v práci (locative) – at work / in work

So V práci literally means In (the) work, but is best translated as At work.

Both are grammatically correct; the difference is in emphasis and flow.

• V práci mám nového kolegu…
  Neutral, common. It sets the place first: At work, I have a new colleague…

• Mám nového kolegu v práci…
  Slightly more focus on the fact that you have a new colleague, and then you add at work.

Czech word order is flexible because meaning is carried by endings. The first position often holds what you want to set as background or topic (here: V práci).

Czech normally omits subject pronouns when the verb ending already shows the person:

• mám = I have (1st person singular)
• máš = you have
• má = he/she/it has

So Mám nového kolegu already means I have a new colleague.
You only say Já mám… when you want to emphasize I (contrast), e.g. Já mám nového kolegu, ale on ne. – I have a new colleague, but he doesn’t.

Který is a relative pronoun, equivalent to English who / that / which.
It refers back to kolegu and introduces a relative clause:

• nového kolegu, který plánuje velký projekt
  = a new colleague who is planning a big project

Který agrees with kolega/kolegu in:

• gender: masculine
• number: singular
• case: nominative (it is the subject of plánuje)

So we use který (masc. sing. nominative), not kterého, kterému, etc.

In Czech, a comma is normally used before a relative clause introduced by words like který, že, protože, když, etc.

Here, který plánuje velký projekt is a separate clause that gives extra information about kolegu. So you write:

• V práci mám nového kolegu, který plánuje velký projekt.

This is similar to English: I have a new colleague, who is planning a big project.

Plánuje is the present tense of plánovat (to plan), 3rd person singular.

Czech has only one present form, which can correspond to both English:

• plans (simple present)
• is planning (present continuous)

So který plánuje velký projekt can be translated as:

• who plans a big project
• who is planning a big project

Context decides which English form is better.

Here, projekt is the direct object of plánuje, so it is in the accusative case.

For masculine inanimate nouns like projekt, the accusative form is the same as the nominative:

• nominative: projekt
• accusative: projekt

The adjective velký follows the same pattern:

• nominative: velký projekt
• accusative: velký projekt

Forms like velkého projektu would be genitive, not correct after plánuje in this meaning.

Velký can mean both physically large and big/major/important.
In the sentence který plánuje velký projekt, without more context it usually suggests:

• a major / large-scale project (big in scope/importance), not physically big.

Context would clarify if it’s about physical size.

No, those would mean something different:

• v práci – at your workplace / at work (correct here)
• u práce – literally by work, usually not used this way
• na práci – for work / for doing work (e.g. mám hodně práce na zítra – I have a lot of work for tomorrow)

For at work (location, job place), you normally use v práci.

Then kolega would be in the nominative, not accusative:

• V práci nový kolega plánuje velký projekt.
  At work, a new colleague is planning a big project.

Compare:

• Mám nového kolegu (accusative: object – I have a new colleague)
• Nový kolega plánuje… (nominative: subject – A new colleague is planning…)

Práce is a feminine noun. Its locative singular form (used after v for location) is práci:

• Nominative: práce – work
• Locative: v práci – at work

So the change -e → -i here is just regular declension: práce → práci in the locative.