I don't know how to transfer over issues as I have not had to do that yet, but passing over the wiki is not that difficult.
On your old gitlab instance you will notice two repositories for your project (let's pretend your wiki is oldproject), one will say something like oldproject.git and oldproject.wiki.git.
The general path to the repositories where you can see the names I am talking about (let's assume user-name is "myaccount") can be found here:
/home/git/repositories/myaccount/
or (if using the omnibus installer):
/var/opt/gitlab/git-data/repositories/myaccount/
I presume you already know how to transfer over oldproject.git. You do the exact same thing with the wiki, only you create a bundle file out of oldproject.wiki.git:
git clone http://gitlab-instance-ip/user-name/oldproject.wiki.git
cd oldproject.wiki
git bundle create oldproject-wiki.bundle --all
Now initialize your new project in gitlab...I presume you already know how to do that as you suggested in your question that you know how to import the files from your project over to the new instance without problem. Now repeat for the wiki:
git clone http://new-gitlab-ip/user-name/newproject.wiki.git
cd newproject.wiki
git pull /path/to/oldproject-wiki.bundle
git push -u origin master
I had a very similar problem to yours where I didn't see that anything was actually "pushed". When I went back to the gitlab project I noticed that it was in fact updated with the wiki. See here if you think it will help: Importing Gitlab Wiki to a new Gitlab Instance
Good luck!