Велосипедист їде до веломайстерні.

Because Ukrainian distinguishes different kinds of motion.

• йти / іти = to go on foot
• їхати = to go by vehicle or while riding something

A cyclist is moving by bicycle, so їде is the natural choice.

So:

• Велосипедист йде... would sound wrong, because that would mean the cyclist is going on foot.
• Велосипедист їде... means the cyclist is riding.

Їде is the 3rd person singular form of їхати.

That means it matches a singular subject such as велосипедист.

You can think of it as:

• я їду = I am going / riding
• ти їдеш = you are going / riding
• він / вона їде = he / she is going / riding

In this sentence, велосипедист is singular, so їде is the correct form.

Not by itself. Їхати is a general verb meaning to go by some vehicle or while riding something.

Depending on context, it can be used for:

• a car
• a bus
• a train
• a bicycle
• even horseback in some contexts

In this sentence, we know it means riding a bicycle because the subject is велосипедист.

So the idea comes from the whole sentence, not from їде alone.

After the preposition до, Ukrainian normally uses the genitive case.

The dictionary form is:

• веломайстерня = bicycle repair shop / bike workshop

After до, it becomes:

• до веломайстерні

So this is:

• до
  • genitive

That is why you do not see веломайстерня in its basic form here.

До usually means to, toward, or to the place of.

So їде до веломайстерні means the cyclist is going to the bike repair shop as a destination.

Using в / у would focus more on movement into the place:

• їде в веломайстерню / у веломайстерню = is riding into the repair shop

In many contexts, both can be possible, but they are not identical in nuance:

• до веломайстерні = toward / to that destination
• в / у веломайстерню = into the inside of the shop

Ukrainian has no articles.

So велосипедист can mean:

• a cyclist
• the cyclist

and веломайстерня / веломайстерні can mean:

• a bike repair shop
• the bike repair shop

Context tells you which meaning is intended.

This is very normal in Ukrainian, and learners have to get used to relying on context instead of articles.

Велосипедист is in the nominative case, because it is the subject of the sentence.

It is the person doing the action:

• Велосипедист = the cyclist
• їде = is riding / going
• до веломайстерні = to the bike repair shop

So the basic sentence pattern is:

• subject
  • verb
    • destination

No, Ukrainian word order is fairly flexible.

The neutral, straightforward order here is:

• Велосипедист їде до веломайстерні.

But other orders are possible if you want a different emphasis, for example:

• До веломайстерні їде велосипедист.
  This puts more focus on the destination.
• Їде велосипедист до веломайстерні.
  This can sound more literary or context-driven.

So the original sentence is the most neutral and natural for a simple statement.

Yes, it is basically a compound-style word.

It contains:

• вело- = bicycle-related
• майстерня = workshop / repair shop

So веломайстерня is a bike workshop or bicycle repair shop.

You may also see related expressions such as:

• велосипедна майстерня = bicycle workshop

Both are understandable, but веломайстерня is a compact, natural word.

The letter ї is pronounced roughly like yi.

So їде sounds approximately like YI-de.

A useful point for English speakers:

• і is like ee
• ї is like yi / yee with a clear y sound at the start

So:

• іде would sound different from їде
• їде begins with that clear y glide

This letter is very important in Ukrainian pronunciation and spelling.

Yes, you could, and it is grammatically fine.

That would make the means of transport completely explicit:

• на велосипеді = by bicycle / on a bicycle

But in the original sentence, that information is already obvious from велосипедист, so adding на велосипеді is usually unnecessary.

So:

• Велосипедист їде до веломайстерні. = natural, concise
• Велосипедист їде на велосипеді до веломайстерні. = correct, but more explicit than needed