A lista szerint kenyérre, tejre és tojásra is szükségem van.

It means according to the list.

• a lista = the list
• szerint = according to

So a lista szerint is a very common Hungarian expression for giving a source or point of reference:

• A szabályok szerint = according to the rules
• A hírek szerint = according to the news

Because after szükségem van (I need), Hungarian uses the suffix -ra/-re on the thing needed.

So:

• kenyér → kenyérre
• tej → tejre
• tojás → tojásra

This is just how the verb-like expression works:

• szükségem van valamire = I need something
• literally, something like I have need for something

So Hungarian does not use a direct object here the way English does.

This is due to vowel harmony, a basic feature of Hungarian.

• Words with mostly front vowels usually take -re
• Words with mostly back vowels usually take -ra

In this sentence:

• kenyér has front vowels, so → kenyérre
• tej has front vowels, so → tejre
• tojás has back vowels, so → tojásra

This same pattern appears in many other suffixes in Hungarian.

Because the noun already ends in r (kenyér), and the suffix is -re.

So when they come together:

• kenyér + re → kenyérre

Both r sounds are kept in spelling, which gives rr.

It means I need.

But grammatically, it is built from:

• szükség = need
• -em = my
• van = is / exists

So szükségem van literally means something like I have a need or there is my need.

This pattern changes by person:

• szükségem van = I need
• szükséged van = you need
• szüksége van = he/she needs
• szükségünk van = we need

Because szüksége van does not take a direct object. It requires the -ra/-re form instead.

So:

• Kenyeret veszek. = I’m buying bread.
  Here bread is a direct object, so accusative is used.
• Kenyérre van szükségem. = I need bread.
  Here bread is the complement of szükségem van, so Hungarian uses -re.

This is one of those patterns you simply need to learn as part of the expression.

is usually means also / too / as well.

In this sentence, it adds the idea that these items are also needed. So the sense is something like:

• According to the list, I also need bread, milk, and eggs.

Even though is is attached to the last item (tojásra is), it can apply to the whole list:

• kenyérre, tejre és tojásra is

That placement is very normal in Hungarian.

Hungarian often uses the singular where English might use a plural in a general sense.

So tojásra can mean egg or eggs, depending on context. Here it is understood as a general shopping-list item.

If you wanted to be more explicitly plural, you could say:

• tojásokra

But in many everyday situations, Hungarian prefers the singular for this kind of general reference.

Because the sentence is talking about the items in a general, indefinite way:

• bread
• milk
• eggs

Hungarian often leaves out the article in cases like this, especially in lists or when naming things needed, bought, eaten, etc.

Compare:

• Kenyérre van szükségem. = I need bread.
• A kenyérre van szükségem. = I need the bread.
  This would refer to a specific bread.

No, Hungarian word order is fairly flexible, and changes in order often affect emphasis rather than basic meaning.

This sentence:

• A lista szerint kenyérre, tejre és tojásra is szükségem van.

is natural and highlights the needed items nicely.

A different order can also be possible, for example:

• A lista szerint szükségem van kenyérre, tejre és tojásra is.

The meaning stays very similar, but the emphasis shifts slightly. Hungarian often puts the most important or most relevant information in a prominent position.

Sometimes, but not in exactly the same way.

• szükségem van valamire = I need something
• kell valami = something is needed / something is necessary
• kell nekem valami can sometimes mean I need something, but it often sounds a bit different in tone or structure

For shopping-list style statements, szükségem van is very natural:

• Kenyérre van szükségem. = I need bread.

With kell, you would usually structure it differently:

• Kell kenyér, tej és tojás. = Bread, milk, and eggs are needed / We need bread, milk, and eggs.

So both can be possible in some contexts, but szükségem van is the more direct match for I need.