Learn Hardcore Portuguese (Portugal): Eu gosto de rir com os meus amigos.

We also do not say gosto do rir here because rir is a verb (in the infinitive), not a noun.
You only use do / da / dos / das when de contracts with a following definite article:

Eu gosto do café. – I like the coffee. (de + o → do)
Eu gosto de rir. – I like to laugh. (no article, so no contraction)

Can I say gosto de rir-me or gosto de me rir instead of gosto de rir?

In European Portuguese, rir-se (reflexive) is very common, and both of these are grammatically possible:

Eu gosto de me rir com os meus amigos.
Eu gosto de rir-me com os meus amigos.

The meaning is practically the same as gosto de rir, especially in a neutral context like this: I like laughing with my friends.

Notes:

Position of me:
- After the infinitive: rir-me (more traditional/European)
- Before the infinitive: me rir (also used, especially in speech)
When you say rir-se de alguém, it usually means to laugh at someone (make fun of), not just laugh near them. With com, it’s simply laughing together:
- rir com alguém = laugh with someone
- rir-se de alguém = laugh at someone

Is eu necessary here, or can I drop it?

You can drop eu in European Portuguese:

Eu gosto de rir com os meus amigos.
Gosto de rir com os meus amigos.

Both are correct and natural.

Because the verb ending -o in gosto already shows that the subject is eu (I), the pronoun is not needed for the meaning.
Eu is used mainly for emphasis or contrast:

Eu gosto de rir, mas ele não gosta. – I like to laugh, but he doesn’t.

What exactly does rir mean here, and how is it different from sorrir?

rir = to laugh (out loud, or laughing in general)
sorrir = to smile

In this sentence:

Eu gosto de rir com os meus amigos.
→ I like laughing with my friends (making actual laughing sounds, having fun).

If you said:

Eu gosto de sorrir para os meus amigos.
→ I like smiling at my friends.

So rir is stronger and noisier; sorrir is more about facial expression.

Why is it com os meus amigos and not com meus amigos?

In European Portuguese, it is standard to use the definite article (o / a / os / as) before possessive adjectives:

o meu amigo – my (male) friend
a minha amiga – my (female) friend
os meus amigos – my (male or mixed) friends
as minhas amigas – my (female) friends

So:

com os meus amigos is the normal European Portuguese form.

In Brazilian Portuguese, people very often drop the article and say com meus amigos, but in Portugal com os meus amigos sounds more natural and more complete.

Why meus and not minhas or meu?

The possessive must agree in gender and number with the noun it refers to:

amigo – masculine singular → meu amigo
amiga – feminine singular → minha amiga
amigos – masculine plural (or mixed group) → meus amigos
amigas – feminine plural → minhas amigas

Here we have amigos (plural, masculine or mixed), so we use meus:

os meus amigos – my friends (all male, or male + female).

Can I leave out meus and just say com os amigos?

Yes, but the meaning changes slightly.

com os meus amigos = with my friends (specifically your friends)
com os amigos = with the friends (context-dependent; could mean our friends, the friends, or just friends in general depending on the situation)

If you want to clearly say my friends, keep meus:

Eu gosto de rir com os meus amigos. – I like to laugh with my friends.

Can I change the word order in this sentence?

The most natural order is the original:

Eu gosto de rir com os meus amigos.

You can move the com os meus amigos part, but you should keep it close to rir:

Eu gosto de, com os meus amigos, rir. – Possible, but sounds a bit formal or written.
Com os meus amigos, eu gosto de rir. – Also correct; sounds like emphasis on with my friends.

Avoid splitting gosto de rir too much in normal speech. The safest, most natural version is still:

(Eu) gosto de rir com os meus amigos.

How do I make this sentence negative: I don't like to laugh with my friends?

In Portuguese, you normally place não before the verb:

Eu não gosto de rir com os meus amigos.
Or, without eu: Não gosto de rir com os meus amigos.

Structure:
(Eu) + não + gosto + de rir + com os meus amigos.

How would I say I like laughing at my friends instead of with my friends?

In Portuguese, laugh at someone is usually:

rir-se de alguém – to laugh at someone
or gozar com alguém – to make fun of someone

More natural options:

Using rir-se de:
- Eu gosto de me rir dos meus amigos.
- Eu gosto de rir-me dos meus amigos.
Using gozar com (very common in Portugal):
- Eu gosto de gozar com os meus amigos.

Be careful:

rir com os meus amigos = laugh with my friends (together, in a friendly way)
rir(-se) dos meus amigos or gozar com os meus amigos = laugh at my friends / make fun of my friends

Is this sentence the same in Brazilian Portuguese?

Very close, but there are some usual differences in style:

European Portuguese (more typical):
- Eu gosto de rir com os meus amigos.
Brazilian Portuguese (very natural):
- Eu gosto de rir com meus amigos. (article often dropped)
- Pronunciation is also different.

Both sides of the Atlantic will understand both versions, but:

Article before possessive (os meus) is more characteristic of European Portuguese.
Dropping the article (meus amigos) is more typical in Brazilian Portuguese.

How do you pronounce the main words in European Portuguese?

Approximate European Portuguese pronunciation (very rough English hints):

Eu → /ew/
- Like English ehw, one syllable: “ehw”.
gosto → /ˈɡɔʃtu/
- go = like gosh without h: gósh
- sto = very short too → GÓSH-tu
de → /d(ɨ)/
- Very reduced, almost like weak d
  - a very short sound; often close to just d.
rir → /ʁiɾ/
- Initial r is a guttural sound (like French r or a soft h): something like heer.
com → /kõ/
- Nasal õ, something like kong but shorter (without fully pronouncing the g).
os → /uʃ/
- Like oosh.
meus → /meuʃ/
- Roughly meh-oosh in one glide: meush.
amigos → /ɐˈmiɣuʃ/
- a = very short, like an unstressed uh → ɐ
- mi = mee
- gos = goosh → uh-MEE-goosh.

Spoken at normal speed, many vowels are very reduced, especially de, os, and the first a in amigos.