Je voudrais plutôt rester à la maison ce soir.

Je voudrais is softer and more polite than je veux.

• Je veux rester... = I want to stay...
  This can sound strong, direct, or blunt.
• Je voudrais rester... = I would like to stay...
  This sounds more polite, tentative, or considerate.

French often uses the conditional like this to make wishes and requests sound less direct.

Voudrais is the present conditional of vouloir.

In this sentence:

• je = I
• voudrais = would like / would want

So je voudrais literally corresponds to I would like or I would want, but in normal English it is usually just translated as I’d like.

Because after vouloir / voudrais, French uses a second verb in the infinitive.

So:

• Je voudrais rester = I would like to stay
• Je voudrais manger = I would like to eat
• Je voudrais partir = I would like to leave

There is no preposition between voudrais and rester.
English learners sometimes expect something like to stay, but in French the infinitive comes directly after voudrais.

Here plutôt means something like rather or instead.

So the sentence suggests a preference:

• Je voudrais plutôt rester à la maison ce soir. = I’d rather stay home tonight.

It often implies a contrast, even if the alternative is not stated out loud. For example, the idea might be:

• rather than go out
• rather than do something else
• instead of the other plan

It is placed before the infinitive because it modifies the action rester.

• Je voudrais plutôt rester... = I would rather stay...

This is the most natural placement here.

If you move plutôt, the emphasis can shift slightly, and some versions may sound less natural depending on context. For this sentence, plutôt rester is the standard and safest order for learners.

Because maison is a feminine singular noun.

• à + la maison = à la maison
• au is used for à + le
• aux is used for à + les

So:

• à la maison = correct
• au maison = incorrect

Also, à la maison is a very common idiomatic way to say at home or home.

Yes, often you can.

• rester à la maison = stay at home
• rester chez moi = stay at my place / stay home

Both are natural, but there is a small nuance:

• à la maison focuses on home/the house
• chez moi focuses more on my place

In many everyday situations, they are very close in meaning.

Because soir is masculine singular, so the correct demonstrative adjective is ce.

• ce soir = this evening / tonight
• cette is used with feminine singular nouns

Compare:

• ce livre = this book
• cette maison = this house

So:

• ce soir = correct
• cette soir = incorrect

This is a very common question.

• soir usually refers to the evening as a time period
• soirée often refers to the evening as an event, occasion, or the experience of the evening

So:

• ce soir = tonight / this evening
• bonne soirée = have a nice evening
• une soirée entre amis = an evening with friends / a get-together

In your sentence, ce soir is used because it refers to when the speaker wants to stay home.

Yes, absolutely.

• Je voudrais plutôt rester à la maison ce soir.
• Je préférerais rester à la maison ce soir.

Both are natural, but the nuance is a little different:

• je voudrais plutôt = I would rather
• je préférerais = I would prefer

Je préférerais states the preference more directly.
Je voudrais plutôt can sound a bit softer or more conversational in some contexts.

A careful pronunciation would be roughly:

Je voudrais plutôt rester à la maison ce soir
IPA: /ʒə vudʁɛ ply.to ʁɛs.te a la mɛ.zɔ̃ sə swaʁ/

A few helpful notes:

• je often sounds like zhuh
• voudrais ends with an è sound: voo-dray
• plutôt = ploo-to
• rester = res-tay
• maison has a nasal ending: meh-zon (without a strong final n sound)
• soir sounds roughly like swahr

In natural speech, je may be reduced slightly, but learners should first aim for a clear pronunciation.