Sold our house earlier this year and now living in late parents house until our newbuild is ready so had lots of experience of packing and dealing with removal firms. Had the second round of removal firms around for quotes this week and while they were all very complimentary of what we have done they have all regaled us with horror stories of what many people do (bags, shrink-wrap, newspaper, thin cardboard boxes full of heavy / fragile items etc etc etc........).
Removals companies will pack for you but you will pay extra, personally I preferred to do it myself as that way I knew exactly what was going where plus 80% of our possessions have remained boxed up while we are living in the temporary house. Also it means that when in our new house there is no rush to unpack. To help I clearly wrote on each & every box with a large black marker (on at least two sides) the basic contents, floor & room. In addition they were all marked with different colour gaffa-tape for my own reference, ones marked with red not being required in the temporary house and even when in our new house not until we are settled in (saves having to read the contents, a quick glance at the tape shows whether it is to be opened or not).
Cardboard boxes. You want twin-wall which can be stacked on top of one another. Plenty of firms on Ebay selling used boxes (they bring their products in from overseas and then sell the empty boxes, very green and gives you great value for money). Various sizes available but generally they are about £10 for ten (inc. delivery) so in no way expensive. Various sizes available. For books and other V.heavy items I used the smaller boxes keeping the larger ones for lighter or bulky items. When you have moved, boxes can be passed on to someone else who is moving (local FB groups are good for this).
Furniture. They will wrap & protect. With chests of drawers etc leave clothes etc in the drawers as they just move the whole item complete.
For wrapping you can used scrunched newspaper but also bubble-wrap is ideal. I buy it in 100M rolls (about £10 each inc. delivery) and in total used over 1KM! No waste as it all gets recovered and will be reused by me or passed onto someone else who is moving.
I also bought a large number (i.e. over 70!) of Tote attached-lid stackable plastic storage boxes (56L & 80L). Some were bought S/H (ex-shop items) and are ideal for items from my garage & workshop while others are brand-new and are used for very fragile items (glassware etc). When we have finally unpacked the older boxes will be sold (plenty of takers out there) and the new / almost new ones will be used for storage.
Pictures & mirrors. Wrapped in bubblewrap and then sandwiched in large cardboard boxes (flat of course), that way each picture has at least two twin-wall layers of cardboard on each side and the overlap at the edges means even if it were dropped the cardboard will absorb the impact long before getting to the edge of the frame.
For materials Ebay really is your friend. Bubblewrap & boxes mentioned above, brown parcel-tape (12-rolls for £8 inc. delivery), fragile tape (6-rolls for £7 inc. delivery), thick black indelible markers (12-markers for £3 inc. delivery).