Vezmu vodítko a venčím štěně.

Vezmu is the 1st person singular form of vzít, meaning to take.

It looks different because vzít is an irregular verb. Its present/future stem is vezm-, so you get forms like:

• vezmu = I take / I will take
• vezmeš = you take / you will take
• vezme = he, she, it takes / will take

So vzít → vezmu is just an irregular pattern you need to learn.

This is a question of aspect, which is very important in Czech.

• brát / beru is imperfective: ongoing, repeated, habitual
• vzít / vezmu is perfective: a single completed act

So:

• beru vodítko suggests I am taking / I take regularly
• vezmu vodítko suggests I’ll take / I’ll pick up the leash once

In this sentence, vezmu focuses on the action as one complete act: taking the leash.

Venčím comes from venčit, which means to walk a dog / take an animal outside, especially so it can get exercise or relieve itself.

So venčit is not a general verb for to walk. It is specifically used for taking a pet out, especially a dog.

Examples:

• Venčím psa. = I’m walking the dog.
• Venčíme štěně. = We’re walking the puppy.

In your sentence, venčím štěně means I walk / am walking the puppy.

Yes, they are direct objects, so they are in the accusative case.

The reason they look unchanged is that both nouns are neuter singular, and for many neuter nouns the accusative singular is identical to the nominative singular.

So:

• vodítko: nominative vodítko, accusative vodítko
• štěně: nominative štěně, accusative štěně

That is why the forms do not visibly change here.

Both are neuter nouns.

You can often recognize neuter nouns by endings like:

• -o → vodítko
• -ě → štěně

This matters because gender affects adjective forms, pronouns, and case endings.

For example:

• to vodítko = that leash
• to štěně = that puppy

Czech normally does not use articles.

So vodítko can mean:

• a leash
• the leash

And štěně can mean:

• a puppy
• the puppy

The exact meaning depends on context. This is completely normal in Czech.

Because Czech usually drops subject pronouns when they are not needed.

The verb endings already show who is doing the action:

• vezmu = I will take
• venčím = I walk / am walking

So já is unnecessary unless you want emphasis or contrast.

For example:

• Já vezmu vodítko. = I’ll take the leash.
  This sounds more emphatic, as if contrasting with someone else.

Yes, absolutely.

Adding si often gives the sense of taking something for oneself / with oneself.

So:

• Vezmu vodítko = I’ll take the leash
• Vezmu si vodítko = I’ll take the leash with me / for my use

With everyday objects, si is often very natural in Czech. In this kind of situation, many speakers might actually prefer vezmu si vodítko.

It is understandable, but a learner may notice a small aspect/tense mismatch.

• vezmu is a perfective form, which usually has future meaning
• venčím is an imperfective present form

So the sentence mixes a future-like verb with a present-tense verb.

A native speaker will still understand it, but for a straightforward future idea, something like these may sound more natural:

• Vezmu si vodítko a půjdu venčit štěně.
• Vezmu si vodítko a vyvenčím štěně.

The original sentence is still useful for learning vocabulary and grammar, but in real-life speech you may often hear one of the alternatives above.

Yes. Czech word order is fairly flexible.

The sentence Vezmu vodítko a venčím štěně has a fairly neutral structure. But you can move words around to change emphasis or style, not the basic meaning.

For example:

• Vodítko vezmu a venčím štěně.
  This puts more emphasis on vodítko.

• Vezmu vodítko a štěně venčím.
  This gives extra emphasis to štěně.

Also, a simply means and here, joining the two actions.