The living room is where you will be spending most of your time and also entertaining guests. You will probably end up spending the most on your living room decor. Of course you need a sofa before anything else. Place a nice coffee or accent table. Put a lamp stand and planters somewhere to accentuate the space. Books will draw the wavering eyes until you completely fill the space. Eventually enhance the room with art, rugs, fixtures and more.
The kitchen might look the lousiest if not attended well. Apart from the necessary utensils, install a dinner table set around the kitchen. If there is no room for a table that large try a small breakfast table. It will serve the purpose and look attractively cute. If there is an island counter, just invest in high chairs. You can try some hands-on décor items here too, small show pieces or wall hanging art for instance. A flower vase can also do the trick.
Bedroom should be cozy, so don’t just lay a mattress on the floor. Do invest in a bed frame and warm covers. A night stand topped with a lampshade will help. Maybe some aromatic candles too. If you do not have wardrobe or cupboard, and especially if it’s a rental, consider a portable storage. Curtains are a must. Block the unwanted light, establish your privacy.
You need a shelf in your bathroom, if it is not already there. This cannot be left to later. Bath towels and linens need investing and so does a vanity mirror; perhaps a storage cabinet too. Bath mats are more essential than they might seem to be.
Everything else will follow, but we cannot be more dependent on these basic essentials.