Most users find it very easy to sort and find contacts when the contacts are segregated by groups. However, it can get daunting when you want to merge contacts across different contact groups or when you want to merge an entire group into another.
By default, you can't do this on the iPhone. In fact, creating/editing/deleting contact groups is a chore (and many times an impossibility) on your iPhone. You can create groups on iCloud.com but that's about it. On a Mac, you can use the Address Book. On a PC, third-party softwares come to the rescue.
Here's the no-brainer way of merging contacts between groups using your iPhone only:
#1. In Settings → Mail, Contacts, Calendars, under CONTACTS select the Default Account to where you want new contacts to go. When you tap the Default Account, you will be shown the contact groups. You will have to select the right group (your default choice).
#2. Next up, email yourself the contacts from the other group that you want to merge into the default group. You can do this by opening the contact (from the Phone app) and tapping on Share Contact and sending it to yourself (via Email or Message).
#3. Once you've got the contact emailed/messaged, add it to your iPhone. When you import the contact, it gets saved to the default group selected in step #1.
Arguably, this is not the best solution but it works for those five or ten contacts you have in a group. Most important thing to note is that there will be duplicates of the contacts you just added. You can remove the contacts from the old group.
A better solution is to use a software like CopyTrans. This StackExchange thread actually has more solutions but I think CopyTrans is probably the best.
When you use CopyTrans Contacts, you can actually drag and drop contacts between different groups. We won't be going through the whole process but once you run CopyTrans and import your iPhone contacts into it, the groups get reflected too.
You can simply select contacts from a particular group and then drag them all to the group you want to move them to. Once done, sync the iPhone to CopyTrans Contacts and the changes take place instantly.
To most users, a third-party solution works perfectly and is much easier. Also, when you've got a lot of contacts in between groups, this is the best solution.