There are two levels of baby-proofing required. The first, and most important is the safety level. This involves removing access to dangerous things so the child will not be injured. The second is the destruction level, which involves protecting valuables and breakables from inquisitive children.
Beginning with the
first level, this should start before baby comes home. A newborn can't get into much mischief, but there are things that can harm them, and it's good to get into safe habits early. This is not an exhaustive list, there are always other things to consider. Please feel free to add tips if you have them. I have tried to create lists for each age of baby, but these should be done before the baby is at that stage - better proactive than reactive, and it's better to make it impossible for a child to injure themselves, than to try and train a child to stay away.
Baby
Never leave them on a change table or bed unsupervised, even for a short time. It's a good habit to get into, and you can be taken by surprise when they figure out how to roll over.
When filling the bath, finish by running the cold tap for a few seconds, to avoid accidental scalds. Never, ever leave a baby in the bath unattended.
Make sure that the cot mattress fits snugly into the cot. Any gaps at the edges can be fatal for a baby if they happen to roll, and get stuck. Look at the distance between the slats on the cot, or the webbing. It should be small enough that they cannot get their head through. Keep the cot as bare as possible, and don't use pillows or electric blankets.
Consider creating a soft landing around the cot/change table for when baby gets bigger and can climb out, or accidentally falls off.
Take a good look at your family pet. It's your decision how/whether to allow access to the baby, but unsupervised access is a bad idea.
Also consider the ages of others in the household. A young child may want to play with the baby without knowing how to treat them.
Crawler
Get down on the ground and look at each room from your crawler's point of view. What can they reach? Are there things that can form steps up to higher areas? Consider closing off certain rooms, or putting up barriers to stop access. My Mum used to say it's easier to keep baby out than to keep baby in (that is, don't try to keep them in a play pen, just section off the areas of the house they shouldn't access).
Look for tablecloths or dangling items that can be pulled down, or electric cords hanging down. What is on top of the tablecloths can fall onto baby.
Look at the location of your appliances, stereo, TV and video etc. Can they be pulled off the shelf?
Look especially for curtain cords that are looped at the bottom. These can be a strangling hazard. Tie them up high or replace them.
Keep plastic bags up high, away from little ones. Drill breathing holes into any closable containers (toyboxes etc), in case baby gets stuck inside.
Low coffee tables can be a problem if hot drinks are left unattended. Change your habits so that tea and coffee is put down elsewhere.
Go through your lower cupboards, looking for poisons, sharp objects and breakables. Swap their storage spot for somewhere less accessible, and put your pots and pans and tupperware in the most accessible cupboards. Consider child locks if this is not possible.
Throw out any household chemicals that are not used, and consider replacing toxic substances with safe ones. Bicarb and vinegar can be used in place of many cleaning products.
Ensure all standing furniture is secured, especially if there are shelves that can be climbed on. A crawler may not be able to get onto them yet, but they grow fast. Bolt anything to the wall if it poses a risk. Consider the weight of the furniture - a lightweight cupboard can be pulled over by a child who is learning to stand.
Look at your oven, especially older ones - when the door is down, a child's weight may be enough to topple it forward, crushing the child or spilling hot food onto them.
Never leave buckets or basins of water around.
Install safety switches - you can never be too careful, and even crawlers can be fascinated by power points. Consider childproof plugs.
Look for hard edges of furniture, especially at baby's head height. Consider padding them with rubber corners.
Keep all small items (anything that will fit into an old film canister) up high and secure.
Toddler
Look for sources of heat - heater, oven, stove, and make sure that your toddler can't get access to them, and cannot turn them on.
Get down to the ground again, and look at each room from your toddler's point of view. What can they reach? Are there ways they can climb up to areas of danger? Some children learn to drag chairs or items around, and make steps up for themselves.
When buying a stool for a toddler to use, make sure it has a wider base than its height, and it has rubber feet. You don't want it to tip or slip.
Keep toaster and kettle in a cupboard if you can do it.
Make sure any outdoor play areas are totally secure and that fences cannot be climbed (especially pool fences). Some toddlers are amazing little Houdinis.
In addition to these tips, at each stage of development, the best thing to do is to watch your child and see what they get into, then proof it as necessary.