What makes a date suitable
For Griha Pravesh, tradition looks at four things: the nakshatra, the tithi, the weekday and the lunar month.
Favourable nakshatras: Rohini, Mrigashira, Chitra, Anuradha, Revati, Uttara Phalguni, Uttara Ashadha and Uttara Bhadrapada. These are described as sthira (fixed) or mridu (gentle) nakshatras, both suited to settling into a place.
Tithis to prefer: Dwitiya, Tritiya, Panchami, Saptami, Dashami, Ekadashi and Trayodashi. The rikta tithis — Chaturthi, Navami and Chaturdashi — are avoided, as are Amavasya and, for most families, Ashtami.
Weekdays: Monday, Wednesday, Thursday and Friday are preferred. Tuesday is generally avoided for house entry. Saturday is avoided by many families though practice varies by region.
Months: Magha, Phalguna, Vaishakha and Jyeshtha are considered strong. Ashadha, Bhadrapada, Ashwina and Pausha are usually avoided, as is the Chaturmasya period by some traditions.
Time of day
The entry itself should happen in the morning, before Rahu Kalam. Abhijit Muhurta — roughly 48 minutes around solar noon — is a universally acceptable window when no other suitable time is available.
When the ideal date is not available
If you must move in on an unfavourable date for practical reasons, families commonly perform a simple Vastu Shanti and kalasha pooja on the practical date and hold the full Griha Pravesh later on a proper muhurat. Discuss this with your purohit — it is an accepted practice.