How can you sell Team Fortress 2 items instantly in the Steam Market and what's the best/easiest way?

The only method to sell an item instantly is to match the buyer's price to an existing buy request.

For commodity item listings

Let's take a look at an example: a "Name Tag" is sold in fairly high quantities and therefore has a "commodity" item pricing style where no individual instance of the item can be selected out of any others at the same price.

Here we can see that there are currently 390 requests to buy a Name Tag at $0.58, so if you were to sell a Name Tag from your inventory set so that the buyer pays $0.58 or less, then the item will be automatically matched and sold to one of those requests to buy.

As a rule the requests to buy will always be at a lower price than the lowest listings to sell, so selling instantly may not always return the greatest profit, especially considering the portion that Steam/Valve takes from each transaction.

If you're alright potentially waiting for a short period of time my recommendation would be to sell the item at a price within the 'price void' of the buy vs sell price chart:

Selling within this range generally will still result in a quick sale, but of course it won't be instant since someone else needs to actually buy the item.

For non-commodity item listings

For items in which each instance of the item is unique there is no real method to instantly sell the item via the Community Market. For every individual item listed there must be someone to visit the listing later and perform the purchase.

In order to truly instantly sell a non-commodity item you'd need to make use of a third-party service that can exchange the individual item for a series of bulk items such as metal, keys, etc.

Once the non-commodity item has been exchanged for bulk items it would be possible to match those to existing buy orders, but at that point the loss in profit between the automated item trading combined with Valve's cut may make this method undesirable for achieving the best profit return.

Alternative Suggestions

A lot of sales of items take place in exchange for other TF2 items, for which there is no percentage taken by any third party and it's generally a barter system. Sites like Backpack.tf can be used to publish trade listings with costs specified in Keys & Metal.

In general, selling an item 'instantly' or the nearest similarity of 'quickselling' won't result in the best value for the seller, but then again time is often worth something as well.