I don't think there's any harm in logging the user in after registering and get straight to the action, since many sites like Twitter already do this.
Some sites tell their users to:
- Type your details to register.
- Check your email.
- Click the link inside the email.
- Retype your email/username and password.
Doing steps 1 to 3 is already a significant amount of effort for a site you're not familiar with. Making the user retype their email and password seems redundant to the user. In most cases, they just did this.
Furthermore, browsers such as Chrome offer to save the password upon registration anyway.
Where possible, let the user start using your site right after registration, with limitations until they confirm their email.