Quick answer: four things are enough to start — a shaker, a jigger (measuring cup), a bar spoon and a strainer. Everything else can be improvised or bought later.
The shaker is the main tool. For home use, the cobbler type with a built-in strainer in the cap is the most convenient: no separate strainer needed. Temporary substitute — a glass jar with a tight lid.
A jigger is a 25/50 ml measure. A cocktail is proportions: eyeballing never keeps the sour-sweet balance. A measuring spoon works as a substitute, but a jigger is faster.
A long-handled bar spoon is for stirred cocktails and layered shots. A muddler is for mojitos and fresh-berry cocktails; a rolling pin end works in a pinch.
Ice is the most underrated 'tool': you need far more of it than you think. Large cubes melt slower and dilute less. Get 2-3 trays and keep a full stock in the freezer.
What you don't need at the start: 20-piece kits, flaming gadgets, pro stations. Put that money into good ingredients instead — the difference in taste comes from what's in the glass.