There are a number of settings you can set for how the checkout process works and looks. To edit your checkout settings:
Go to your Shopify Admin > Settings > Checkout.
Style - You can edit the way your checkout page looks by clicking the "customize checkout" button, which will pull up the theme customization page, where you can set a logo for the checkout page, as well as the colours used.
Customer Accounts - I recommend setting this to 'optional' for most business types, as it gives customers the ability to check their previous orders and will automatically fill in their address for them on subsequent orders during the checkout process. If you set this to customer accounts are required, then you may lose some potential orders/customers who are either in a rush or are feeling too lazy to make an account. You can also disable this option completely (usually recommended for stores that sell single products, or 1-time services).
Customer Contact - These options are well defined on the page, and are a matter of personal preference. I usually go with "Customers can only check out using email" so that I can get their email for my mailing lists -- and then check the box for receiving shipping updates.
Form Options - The following options are good ones to use, regardless of your business type (they're what I use for every store that I setup). Require first and last name, company name is optional, address line 2 is optional, and shipping address phone number is required.
Tipping - If your business is service-based then you can check this option to offer a box during checkout where the customer can choose a tipping amount.
Order Processing - Definitely check the first two options for, "use the shipping address as the billing address by default," and "enable address auto-completion." Set the 'abandoned carts' setting to "1-hour later" and the 'promotional emails' setting to "by default, customer agrees to receive promotional emails." Unless you're selling exclusively digital products, then you will want to set the 'fulfillment' setting to "automatically fulfill only the gift cards of the order." I usually don't check the box for automatically archiving orders, as I like to manually archive them once I know the customer has actually received their product and has no complaints or issues. For the additional scripts section, you will normally leave this blank, unless you have some tracking codes you need to include (usually from Google Analytics).
Checkout Language - This should always be in the same language as your store (ie. English, if everything on your site is in English). If you're using an app to support translations of your site, then make the checkout language be the language used by the majority of your customers.
Refund, Privacy, Shipping & TOS Statements - You will 100% want to fill these out. Do not leave them blank. If you don't have a statement prepared for each section, you can click the "generate sample policy" button next to each section, and Shopify will create a statement for you. Definitely read through the generated statements, and edit them as needed, so that they're accurate, and you know exactly what you're promising customers. Update: the settings for these have been moved to their own page. Go to your admin > settings > legal ... in order to edit and set these up.
Need more help with this step? Check out Shopify's help docs for checkout settings by clicking here.