watasi ha syuumatu ni atarasii mise ni ikitai desu.

They serve two different functions:

• 週末に marks a time expression (“on the weekend”).
• 店に marks a destination (“to the store”).
  In Japanese you often see multiple にs when you combine time and place/direction.

は is the topic marker. It tells the listener “we’re talking about me.”

• You could also use が as a subject marker, but は emphasizes “as for me …”
• In casual speech, you can drop 私は entirely if context makes the subject clear.

行きたい means “want to go.” It’s the -たい form, which turns a verb into an i-adjective expressing the speaker’s desire.
Formation:

1. Take the verb stem (行く → 行き).
2. Add たい ⇒ 行きたい.
   You can then conjugate it like an i-adjective (e.g. 行きたくない, 行きたかった).

Adding です makes the sentence polite. Since -たい is treated like an i-adjective, you can attach です for extra politeness:

• 行きたいです。 (polite)
• 行きたい。 (plain/informal)
  Neither is grammatically wrong—です just raises the formality level.

Yes. Both に and へ can mark direction/destination:

• 店に行きたい: neutral “I want to go to the store.”
• 店へ行きたい: slightly emphasizes the direction or movement toward the store.
  In practice they’re largely interchangeable in this context.

Yes. Japanese often omits the subject when it’s clear from context.

• 週末に新しい店に行きたいです。
  It’s still perfectly natural; people will infer you’re talking about yourself.

Standard Japanese word order for adverbial phrases is:

1. Time -> 2. Place -> 3. Verb
   So 週末に comes before 新しい店に.
   You can swap if you want, but it may sound less natural:
   
   • 新しい店に週末に行きたいです。

You cannot use -たい to describe someone else’s feelings (that’s for the speaker). Instead you use -たがる, which treats the desire as an observation:

• 彼女は新しい店に行きたがっています。
  Here 行きたがっている indicates “she seems to want to go.”