Open up the Contacts app on your old phone and just scan through the list of your contacts. Now using a web browser (this will be easier on a computer or laptop) log into your Google account Contacts by going to:
https://contacts.google.com
Scan the list of contacts and if it's a match to what you saw on your old phone's Contacts app that's a pretty good sign your phone and your online account were syncing properly. Also check on your old phone in its Settings >> Accounts. Click on 'Google' and then click on your gmail address name. A listing of all the Google services in your Google online account should show up, hopefully most will have green checkmarks including the 'Sync Contacts' box. You could also do a test by just creating a temporary, non-valid contact using either your old phone's Contact app or the web browser online account and that temp contact should pop up in the other. You should be OK as once you set up your new phone and enter your Google account info to activate it, things like your gmail, calendar, and contacts will just transfer over.
In the event your old phone wasn't set up to sync with your online Google account, you need to manually transfer your contacts over from your old phone to the new one. (And you should enable the online syncing as it makes it easier to transfer but more importantly it's an online backup of your contact info.) Start up the Contacts app on your old phone, tap on the menu button, and the should be a 'Import/Export contacts' option. In this case you want to export a file, probably named 'Contacts.vcf'. Once that's done, locate that Contacts.vcf file, and copy it over to your new phone. On your new phone, start up your Contacts app, click on its menu button, select 'Import/Export contacts' and now you want to import that Contacts.vcf file. You might have to sort through things if there are duplicates and such but check that your new phone is syncing with your online Google account.
I'm not familiar with a 'Memo app' and while there are lots of memo type apps on Google Play I don't see one specifically named just 'Memo'. I'm guessing it's a carrier-branded app that's only available for that product line. Anyway, I don't have a clue as to how to transfer saved Memo info. What model is your old phone and what version of Android is it running? Also, is your new phone made by the same company and does it also have this Memo app on it?