Learn Hardcore Romanian: Planul meu este la fel de important ca planul tău.

planul meu = my plan (literally: the plan my)
Saying meu plan is possible only in special, emphatic, or poetic styles; in normal speech you put the possessive after the noun.

So the regular, neutral order is: noun + definite article + possessive
→ planul meu, cartea ta, prietena lui, etc.

What does the -ul at the end of planul mean?

The ending -ul is the definite article for masculine singular nouns – it means “the”.

plan = a plan / plan (indefinite)
planul = the plan (definite)

With a possessive, Romanian often still uses this definite article, even though English does not:

planul meu = literally the my plan, but it just means my plan.

Can I use e instead of este in this sentence?

Yes.

este is the full, neutral form of “is”.
e is the shortened form, used very commonly in speech and in informal writing.

Both are grammatically correct here:

Planul meu este la fel de important ca planul tău.
Planul meu e la fel de important ca planul tău.

In more formal writing, este is slightly preferred, but e is not wrong.

What does the structure la fel de important ca literally mean, and how is it used?

The whole chunk la fel de … ca corresponds to English “as … as”.

la fel = the same / equally
de = a linking word used before adjectives/adverbs in degree expressions
ca = as / like

So la fel de important ca literally means something like equally important as and is used exactly like as important as:

Planul meu este la fel de important ca planul tău.
My plan is as important as your plan.
El este la fel de înalt ca mine.
He is as tall as me.

Why is it ca here and not decât? When do I use ca vs decât in comparisons?

In Romanian comparisons you generally have two patterns:

Equality (as … as)
Use la fel de … ca (or tot atât de … ca):
- Planul meu este la fel de important ca planul tău.
- Este tot atât de rapid ca tine.
Here you do not use decât.
Superiority / inferiority (more/less … than)
Typically use mai … decât (or sometimes mai … ca):
- Planul meu este mai important decât planul tău.
  My plan is more important than your plan.
- El este mai înalt decât mine.

Also, be aware of nu … decât = only:

Nu am decât un plan. = I only have one plan.

So with la fel de, always use ca, not decât.

Why is the adjective important not changing its form here? When would it change?

Romanian adjectives agree in gender and number with the noun they describe.

planul is masculine singular, so the adjective uses the masculine singular form: important.
Feminine singular: importantă
Masculine plural: importanți
Feminine (and neuter) plural: importante

Examples:

Planul meu este important. (m. sg.)
Ideea ta este importantă. (f. sg.)
Planurile noastre sunt importante. (n. pl. treated as f. pl.)

In your sentence, because planul is masculine singular, important stays in its base form.

Why is it tău and not ta or tale after planul?

The possessive adjective must agree with the noun it follows in gender and number.

For tu (you, singular informal), the forms are:

Masculine singular noun: tău
Feminine singular noun: ta
Masculine plural noun: tăi
Feminine / neuter plural noun: tale

So:

planul tău (plan = masc. sg.)
ideea ta (idea = fem. sg.)
prietenii tăi (friends, masc. pl.)
ideile tale (ideas, fem. pl.)

Since planul is masculine singular, you must use tău.

Can I drop the second planul and just say something like Planul meu este la fel de important ca al tău?

Yes, that is perfectly correct and very natural:

Planul meu este la fel de important ca al tău.
= My plan is as important as yours.

Here:

al is a possessive article that stands in for the missing noun (planul) and agrees with it in gender and number.
al tău literally means yours (masc. sg.).

Other patterns:

ca a ta (for a feminine singular noun)
ca ai tăi (masc. pl.)
ca ale tale (fem./neut. pl.)

How would I say this politely to someone I address as dumneavoastră instead of informal tu?

For the polite form dumneavoastră (you, formal), the possessive is dumneavoastră as well, and it does not change for gender or number.

You could say:

Planul meu este la fel de important ca planul dumneavoastră.
My plan is as important as your plan (formal).

Or, dropping the repeated noun:

Planul meu este la fel de important ca al dumneavoastră.
My plan is as important as yours (formal).

How do you pronounce tău, and what sound does ă represent?

tău is pronounced approximately like “tuhw” in English:

t as in top
ă is a schwa sound, like the a in sofa or the e in taken (unstressed “uh”)
u here forms a glide, giving something like a quick w sound at the end

In IPA, tău is /təw/.

The letter ă always represents this short, central “uh” sound in Romanian, and it is different from â/î, which represent a closer central vowel.