This morning, my father invited me to play with the VR game called Beat Saber.

You can say this in three different ways I think:
"to play a VR game called Beat Saber"
"to play the VR game Beat Saber"
"to play Beat Saber"

The first one implies that the person you are speaking to probably hasn't heard of the game.
The second one I think is more neutral as you are not suggesting the other person doesn't know the game already but still giving enough information for them to tell what is is if they don't already know.
If you are sure that the person does know of the game, you would just say "to play Beat Saber".

We would always just say "play the game" or "play X" instead of "play with the game" or "play with X". We only use "play with" if you are talking about playing with someone else or if you are using toys. So you'd say "the kid plays with his toys" or "the little girl plays with dolls", but when talking about a specific game or video games you just say "play X".

He is a freelance programmer, and bought the VR headset, Oculus, to develop a VR chat studio.

I was excited to play with Beat Saber even though I'm a terrible gameat playering games.

To say you are terrible at something, there are two ways:
"a terrible X-er" or "a terrible X-ist"
"terrible at X-ing"

For playing games, if using the first one you can use "gamer" as in "I'm a terrible gamer". This is fine, but the word "gamer" sort of has connotations of someone who plays games all the time and is really good at them so it still sounds a little weird.

I'd usually just say "terrible at playing games". You can also say "terrible at gaming" which is also fine, but again "gamer" and "gaming" have connotations of someone who is really good at playing games and does it often.

I think in general that it is more common to say "terrible at X-ing" than the other way.

This game reminds me of thea popular arcade game in Japan called "Taiko no Tatsujin", which is a series of rhythm games and was also released in North America as a PS2 video game titled "Drum Master".

It is really difficult for people like me who don't have a good taste in moving along withsense of rhythm.

My parents are also poorbad at rhythm games, so they seemed to have already begotten bored with the game.

Using "poor at" is correct but a lot less common than just saying "bad at".

As well as "bored with" or "bored of", you can use:
"sick of"
"tired of"
I think these two examples I gave are a little more commonly used in conversation.
For example,
"I'm so sick of all this homework!"
"I'm getting really tired of you complaining all the time."

Though they assumed that I wouldn't make it through even in an easy mode, so they were surprised that I could clear a hard mode within 20 minutes.

You can use "in hard/easy mode" when you have turned on a setting to make it hard or easy.
You can use "on a hard level" or "on an easy level" if the specific level you're playing is hard/easy but it varies between levels and you haven't actually chosen to always play on hard/easy.

They said that the period of time to baken to become good at something must depend on one's age.

You can also say this without "period of" as in "the time taken to become good at something".