To begin linking all three platforms you need to make sure you are signed in to the correct Google account that has access to each of them.

Merchant Center to Google Ads

Log in to both accounts and make sure you know your Google Ads ID.  This is the 10-digit number in the top right corner of Google Ads (xxx-xxx-xxxx)

Next, log in to your Google Merchant Center and select the spanner icon in the top right corner, then select Linked accounts:

Under Other Google Ads Accounts, select Link Account:

Then enter your 10-digit Google Ads ID and click Send Link Request:

You then need to return to Google Ads and select the cog (tools & settings) > Setup > Linked Accounts:

Scroll down the page until you see Google Merchant Center, and then select Details:

You will then notice your link request from Google Merchant Center.  Select View Details: 

Then select Approve to finish linking your two accounts:

Google Analytics to Google Ads

Open Google Analytics in your browser and log in using the same Google Account you have used for Google Ads and Merchant Center. Navigate to the Admin tab in the bottom left corner of Analytics:

In Admin there are a significant number of options.  You need to select Google Ads Linking in the middle column Property:

You will then see + New Link Group in a red box  Select this tab and you will then be brought to the configure page below:

As you have admin access to your Google Ads account it will automatically be listed:

Select the tick box next to your Google Ads account and then select Continue to name the link.  The link name will be auto-populated as the name on your Google Ads account.  Ensure the link is set to On before selecting Save:

Once this has been saved, all three of your accounts will be linked and you will be ready to go live with a Google Ads Campaign.