Strainers
The strainers are actually nothing more than the sieves you need to pour your cocktail. There are three that bartenders use. Do you need all three at home? It is recommended, but with a Hawthorne and a Fine strainer you can also go a long way, if you do not want to purchase a Julep strainer. Do you really want to practice some more and pour your cocktails at home like a pro? Then I would definitely buy all three.
Hawthorne strainer
The Hawthorne is really one that you must have if you use a Boston shaker or Parisian shaker. It is the sieve that ensures that no ice cubes or large ice pieces are transferred from the shaker into your glass. You can't stop the small ice crystals with it. Then you need the fine strainer. But these ice crystals can be beautiful when they float on your drink.
The Hawthorne has a spring on the side and you place it on your shaker. The spring retains the large pieces of ice and any pieces of fruit you have used.
If you use a cobbler shaker, you do not need this strainer. The cobbler already has a built-in strainer and makes the use of the Hawthorne unnecessary.
Julep Strainer
A Julep strainer looks a bit like a small slotted spoon. It is a wide, flat spoon with holes in it. This strainer is used in combination with a mixing glass. It holds the ice (which in this case does not contain small pieces because you have only carefully stirred it) so that you can pour the drink. You can possibly do this with a Hawthorne strainer. Yes, that is quite possible, especially if you have one that you can place in your mixing glass and not just on it. So if you have to choose between a Julep and a Hawthorne in terms of budget, go for the Hawthorne.
Fine strainer
There's a good chance you already have this strainer at home. A fine strainer (or double stainer) is actually just a kind of tea strainer. You only use this when you shake a cocktail in combination with the Hawthorne. For example, if you use herbs or fruit, the small pieces will come along if you only use the Hawthorne. The fine strainer catches these small pieces and sieves them out.