Items i suggest to the new mums i work for :
Sleeping
Use cot from the start or travel cot with bassinet or carry cot on pram system.
Sleep bags, sheets, odd blanket.If you get fitted sheets for cot they will also be useable on travel cot and carry cot.
Travel
Buy a car seat that fits your car from birth to 4yrs/18kgs. Baby car seats get heavy and many parents find baby is too long for parents comfort, there is a rear facing car seat that is for this age group.
Transport
Pram set up for your lifestyle. With rain cover, sun shade to UV50+ protection, snuggle bag, shopping basket.
A small bag for spare nappies, odd toy, odd growsuit, bottles. The bigger the bag the more you put in and the heavier/bulkier it is when juggling baby, unused bag and shopping.
A carrier/sling.
At Home
Value packs of vests, growsuits. socks, bibs.
Muslins, cardi, hat/mittens/bottees, swaddling wrap (a stretchy jersey blankets works) is nice as it stays wrapped.
Nappies - dispi or cloth & wraps, cotton wool or lots of cheap face cloths, bepanthan / metanium cream, storage box for bedroom and lounge.
Bath/Room thermometer, soft sponge, olive oil to help dry skin on wrists/ankles, scalp.
Nail clippers/scissors.
Feeding
Breast pads, express machine, nipple cream, nipple sheilds.
Bottles, bottle brush, steriliser (if you want or sterilising bags for microwave)
Soother/dummy.
Family towels are fine as you will often need a new towel per bath due to soiling
Toys/bouncy chairs/activity centres/swings are optional.
Looking at local papers in for sale listings for great bargains of short term usage items like bouncers etc.