This article describes how to transfer an account from Robinhood to Fidelity.
It is better to transfer the account than closing positions, since you will avoid the taxable events. Also, you will stay in the market during the transfer.

Transfer money from Robinhood

Click Transfer

On the next screen, choose Transfer an account to Fidelity and click Continue

Type Robinhood in the box and click Next

You will need your Robinhood account number on the next screen. You can find this number on the Robinhood web interface by going Account, Settings.

In the Account screen, you will see a 9 digits Account Number (RHS)

Enter the Account Number in the Fidelity Account number box. Choose the type of account, in my case, Individual.

Click on Next

Choose if you would like to transfer to an existing account. If you already have a Fidelity account, I suggest using an existing account, but do however as you please.

On the next screen, you will choose to perform a full transfer by selecting Yes so you can close your Robinhood account and click Next.

You will get a confirmation screen. Verify all the details and click on Submit

You will get a confirmation screen with all the details. I suggest you print this page for your record.

You are done, congrats! Now you just have to wait a couple of business days until the account transfer is completed.

On the next business day, you will receive an email from Fidelity confirming the transfer:

And the day after (on a Saturday in my case), you will receive an email from Robinhood confirming they have received the transfer request:

Details

While the transfer is going, Robinhood will cancel any pending orders you have. You will also be unable to close any open positions during the transfer, so make sure you are fine holding those securities for the duration of the transfer period. To be on the safe side, I expect this process to take at least 2 weeks (they mention 5-7 business days).

Note: It only took 3 business days for the money to be available within Fidelity.