Your purchase history will show alongside payments to friends and family in your personal transactions feed (the “$☰” icon to the right of your friends feed). “Your stories” will give you the option to share a purchase with others (purchases are private by default), and “Transactions” will provide transaction details and the amount of money in your account after completed purchases.
Additionally, you can always view your Venmo statement by visiting venmo.com on the web, then clicking on Statement. This information can be viewed on the site or can be downloaded as a CSV by clicking on the “Download CSV” button next to the view button.