Chandigarh- Rock Garden, Sukhna Lake.
Patiala- Sheesh Mahal
Amritsar- Jalianwala bagh, Golden temple, Wagah Border. Do experience Love in Amritsari Naan from Amritsar :)
If Amritsar is on your list then I'll suggest you stay at any hostel.. I will recommend GoStops Amritsar) it'll cost you not more than 200 for an accomodation. but it is great if you're solo. the place is clean usually and Is near the Golden temple.