You will notice early on in London Life that some buildings won't let you enter if your outfit's formality isn't high enough. It shouldn't take long for you to accumulate enough wealth to afford fancy enough clothing to get into these places. Also, clothing affects your Coolness and Impression, which has various effects in the game.
List of Clothing
Go to the list of clothing to see all available clothing items in London Life.
Shopping for Clothes
There are several places in Little London where you can buy new clothes. Chess Department Store is one of the first places that you have access to, but there are other shops, so be sure to talk to everybody to find them.
When you select a clothing item in a shop, you will see three bars below the image of the clothing. The top one, C, stands for Coolness. The middle one, I, stands for Impression. The bottom one, F, stands for Formality.
Changing your Outfit
To change your outfit, press X, then choose Outfit. You can select clothing items from the list. The items in grey are the ones that you are already wearing. The items in black are items that you are not wearing. To remove an article of clothing that you are wearing, just choose it in the list of clothing.
Coolness affects how well you can handle negative experiences. For example, if you are wearing a very cool outfit, you will not lose as much happiness if a bat flies over to you, or if you talk to a member of the Family.
A higher impression will make it more likely that you will receive quests that have large Wealth rewards.
Higher formality lets you access fancier establishments like Clement's, the Crumpet Hotel, the Cruise Bar, and so on.