Large numbers of photos on your hard drive will not slow down your pc; keep them on the hard drive.
Picasa is primarily a photo organiser and editor, although PicasaWeb does store your photos on the web. I would not rely on any such thing as the only storage of my photos.
As you are prepared to change your laptop entirely, can I suggest you reset yours to 'factory default'. This will restore your laptop to exactly the way it was when you bought it - it clears all the rubbish that has gathered over the years.
It is easy to do BUT you do need to back up all your files and photos to an external source first - they will be wiped from your laptop.
Be prepared to put a few hours aside to do this as you will need to reinstall programs that you use, and of course updates.
If you are interested in doing this, ask again and you'll get all the info you need.