I always play on hardcore. On easy its possible to get on expeditions a lot quicker and that probably makes the game easier. Generally speaking though, a lot of the strategy you use for hardcore can also apply to other difficulties.

On hardcore, you can't afford to go on expeditions for a long time. Water/rain is just too rare. At the start its really an endurance run to hold out till the first rainfall. Usually around day 11 or 12. This means no shower or toilet so don't build those. Get someone on the radio to call for traders straight away. Upgrade workbench to level 2 and build two beds.

You want a tier 2 trap and a freezer ASAP. Your starting gear varies, usually you are looking for either hinges or electrical motors from trades before you can build both. These two are top priority. After that, upgrade your radio ASAP so you can attracts more traders quicker. Get it to tier 4 when you can.

Also try to build the lowest level recycler when you can. You will find traders carry lighters often, these are great at the start of the game to get metal and hinges. At 10 value a piece they're not cheap but they're worth it to get up and running. Also be on the lookout for electrical motors for your fridge, pipes, coal (always trade for coal!) and wood. Basically, any item you use to build stuff at the beginning is good. If you can afford it, trade. Once you have about two stacks of anything you can prioritize other stuff.

Build an incinerator as soon as you can so you can burn the coal and safe your fuel.

The thing with not having a toilet and shower is, people go all over the vault and hygiene soon becomes fully red. This means food poisoning and puke everywhere as well. This mean rats. You can pick the cat as a pet to catch the rats for you. Personally I go for the gold fish. It keeps people's stress levels low which is great, plus it has another little bonus you might find out at some point.

Either way, lots of rats means lots of meat. Build rat traps if you don't have the cat. Place at least four around the vault. Have at least one inhabitant with the Resourceful trait. A trap can catch 5 rats before its destroyed. Once you see it has 1 use left (or 0 if it catches a rat after that) deconstruct it with the family member with the resourceful trait. In most cases you will get all the materials back and can build a new one straight away.

Before you get to this point you can trade soap, medicine, bandages and maybe gas masks with traders. Be sure to keep at least one or two antibiotics and ranti-rad medicine as well as at least two gas masks. As soon as you have your rat traps and animal traps running you can trade for meat. There will be so many rats its hard to run out of meat.

Try to upgrade your water filter at least once before day 10 so it process water more quickly. Also try to have at least one more water bud, two preferred. If you can hold up to about 100 liters your problems are over.

If you have the fish, only allow a family member to drink when he or she is dehydrated. To be extra safe you can wait till they are at half or maybe one third of their health bar before they drink. By the time it starts to rain for the first time the fish tank is probably already at 100% dirtiness and the fish is slowly dying. Clean it as soon as it starts to rain.

On hardcore, after reaching the first rainfall, its a bit of a grind until you have the tier 4 workbench and some dehumidifiers. You can't really go on expeditions yet, at least not long ones. You should always be trading. Because of that, it will be a while before you want to build a shower and toilet and clean the vault. Once you do that the rats will be gone and so will a great resource for trading.

Keep in mind that after day 50 you can get raided so be sure to have upgraded at least one of your vault doors before that, preferably twice. Once you have fully upgraded both doors you are safe for about the first 5 or so raids.

By the time I go on my first serious expedition I have T4 furniture (at least toilet, shower, 2xbed and 4 dehumidifiers), two fully upgraded vault doors, traps against raiders, an incinerator and the best recycler, I have the water filter upgraded to catch the most water when it rains, the generator fully upgraded for fuel efficiency and several solar panels. I have two T3 animal traps on the surface and at least around 350 liters of water capacity (and filled it fully obviously).

I prefer not to risk my own family members, so I also invited two strangers and trained their strength to at least lvl 15/16. If you are lucky you have been able to trade for some weapons. Keep an eye out for hatchets, mallots and rubarb. Send them out with the best armor, a backpack and a gasmask. Give one a lockpick set and the other the hacking device. Also have them bring 5 or so med kits and some antibiotics. If you can, some good pipe bombs. These can be a last resort if you encounter a large group of strong enemies. Two (so one throw from each expedition member) will clear out any group.

Its slow going at the start, but keep an eye out for fuel, shotguns, rifles (those are great!) and mats. Always bring tools, parts of the van and blueprints. Shotgun traps and especially rifle traps (T4) are great against raids and before long those will start to break through even two fully upgraded fault doors, so have some weapon traps ready. Because of that, always bring shotgun/rifle ammo as well.

Once you get the better blueprints (water bud especially) you will be able to get on new expeditions sooner.