Not sure where to start here and given all the versions of Android I may get some of this a little off. I sync my wife outlook on the pc to her outlook.com account and from there I sync it to her older Android Samsung S3. I say older as I can't get contact syncing working going from phone (Android 6.0) to outlook.com but I'm working on it. Her old S3 is flawless in both directions for calendar, tasks and contacts.
Before we start, in Outlook for the pc or outlook.com make backups of contacts and calendars using export. These instructions get your contacts and calendar showing in the Android base apps as opposed to Outlook Android app. Can't believe as others have stated that Outlook app can't edit / add / delete contacts. ****
I won't call myself an expert, the choice to follow these instructions is yours.
I saw someone said they don't want mail syncing. So be it, get a new outlook.com and don't give out address, use this account as a sync portal for calendar, contacts and tasks.
First you need to follow the instructions at http://www.slipstick.com/outlook/olc-hotmail/sync-calendar-contact-using-outlook-com/ This gets outlook on PC syncing with Outlook.com. Diane said she would keep this article up to date as outlook.com accounts convert over to the new look which I think they are referencing as an Office 365 account. Thank god, I'm tired of explaining the difference between outlook for the PC and outlook on the web. Contacts are straight forward as CSV files. The calendar process is documented at http://www.outlook-tips.net/outlook.com/export-outlook-calendar-outlook-coms-calendar/ .
Note if you already have outlook syncing with outlook.com you should have got a notice from Microsoft telling you to reconnect your account to outlook on PC. If you don't syncing **** will ensue and mail could be addressed wrong people. Nice. Basically remove account from Outlook for PC and re-install. Here is the link on re-installing.
https://support.office.com/en-us/article/Add-your-Outlook-com-account-in-Outlook-for-Windows-642c1902-bdd9-4dc3-abe7-76d60b148b23?ui=en-US&rs=en-US&ad=US Don't forget to check your default email and data account settings after reconnecting.
Now on to Android. You have to create account as outlined in first post of this thread. I found this from Vodafone. Do not pick Outlook account to add.. http://vodafone.intelliresponse.com/index.jsp?id=1593&question=Set+up+Hotmail%2FOutlook+as+an+Exchange+account+on+your+Android+device&requestType=NormalRequest&source=100 You pick Exchange even though it has Google M on it. The port will be right but as first poster mentioned the server is m.hotmail.com. You will be prompted for passwords and acceptance of permissions to access data. Answer "Yes".
In Android 6.0 tap Settings and scroll down to accounts. Tap on your Exchange account. If you don't see an exchange account review paragraph above. If you see an outlook.com account ignore it for now. When you tap on your exchange account, you may have to scroll down a little bit, you will see sliders beside mail, contacts and calendars. Ensure they are on.
When your account is set up the first task is to get calendar running. Tap your calendar app, with 31 on it. Android 6.0, in the upper left corner there are three bars, the hamburger, tap it. This should show you, as you scroll down, which accounts are eligible. Tap off the Gmail ones and enable your Exchange. If your exchange account is not listed see paragraph above to ensure calendar syncing is on. Run a test. It syncs both way although it can take time going from phone to web. Once you are happy move on to contacts.
Now tap on contacts app. There are three dots upper right corner. I think the option we want is display contacts. Shut off gmail or all and turn on Exchange account. You should see your contacts. Change something in a contact on outlook.com account and it should sync to phone and to Outlook for pc. If outlook for PC does not get change click on the inbox for your outlook.com account within Outlook for the PC program (yes there is another error and when you click on the inbox it actual connects to web account. You only need to do this once each time you start outlook for pc. Sometimes in connects on its own. I just click on the inbox every time I start, it takes a second.)
If you downloaded the Outlook app you may have to go in an shutdown syncing of contacts within the app. It is a one way sync which shows you your contacts, there is no functionality to mod so I would hardly call it syncing, more showing. If you don't do this you will notice that entries for a person in Android contacts may show different results as the syncing is whacked. It seems the app push its contacts over to Android contacts even if you say don't display. If you have your calendar in Android app, contacts in Android app and mail somewhere else you may just want to delete. If enough of us delete MS will get message, fix contact functionality.
This should give you two way syncing for calendar and the ability to change a contact on web or Outlook for PC and the change will appear everywhere. Other versions of Android are similar. In calendar and contacts you have to use the app settings in each app to signify which accounts the apps should use. Apple just turns them all on and if you don't see appropriate defaults and you have multiple accounts syncing **** will surely follow you depending on order you added accounts.
Still working on making sync work on Android 6.0 for contacts. That is how I found this post. I see things like making sure contact or calendar app has permissions, (didn't we already grant when we set up account??), allow data usage in background, reset phone and use outlook.com as default (first) account, ensuring date time is correct and the same across platforms (timezone). One fix was shut off phone and restart it but only the lucky ones get that as a fix.
Hopes this helps a little.