Here's the identification and correction of errors in the sentences provided:

1. **Error:** "Don't forget to put up the rubbish every evening for the garbage collection."
- **Correction:** "Don't forget to put out the rubbish every evening for the garbage collection."
- **Incorrect word:** put up → **put out**

2. **Error:** "I go to the supermarket to shop for groceries at present, so that I can prepare lunch later."
- **Correction:** "I am going to the supermarket to shop for groceries at present, so that I can prepare lunch later."
- **Incorrect tense:** go → **am going**

3. **Error:** "Now, I am preparing dinner for the family while my sister tidies up the living room."
- **Correction:** "Now, I am preparing dinner for the family while my sister tidies the living room."
- **Phrase:** tidies up → **tidies**

4. **Error:** "Now we are gathering around the TV to watch our favorite game show named 'The Amazing Race' every Friday."
- **Correction:** "Now we are gathering around the TV to watch our favorite game show called 'The Amazing Race' every Friday."
- **Incorrect word:** named → **called**

5. **Error:** "I feel so tired at the moment, so I am wanting to have a sleep."
- **Correction:** "I feel so tired at the moment, so I want to have a sleep."
- **Incorrect phrase:** am wanting → **want**

These corrections address the grammatical errors and provide clearer phrases or words.