I have been in Abkhazia via and to Georgia just for 3 days during the last week and I am happy to share my experiences:
Zugdidi to border
You can easily get a taxi from the train station or the Zugdidi main place to the border. We paid 10 Lari without any large negotiations. If you are good in negotiating you might get it slightly cheaper. The drive is approx. 15 minutes to the border.
The border from Georgia to Abkhazia
The taxi dropped me at the border and a Georgian policeman stopped us. He checked our passport and registered that we cross he border. He asked us where we will go and for the purpose. "Sukhumi" (important: Call it Sukhumi here!) and "tourism" did it perfectly. He then asked for our clearance letter to Abkhazia. When we showed it to him he said: "No! I do not want to see it. I am not interested in this puppet governments official papers. I just want to be sure that you have one so you will not be back in 5 minutes!". That's it basically.
Then you do the 1km over the bridge, which was horribly long with my heavy luggage. Horse "cars" go only sometimes and I did not get their "schedule".
The Abkhaz side asked for a first check of the clearance letter and the passport before the real control. Then you have to give both through a "mirrored window" to a border guard. After checking everything for 10 minutes we were waved through.
In Abkhazia
I have been in Abkhazia with a NGO and was invited by the government. So I got a lift from the border to Sukhum and cannot say much about this route or public transport.
In Sukhum itself I stayed in Ritsa hotel, which was very precious, but a bit worn down. Still comfy and absolutely good enough.
Our host, the government, gave us a lot of free time where we walked through Sukhum mainly. Especially when it got late they checked back frequently if all is fine. They also told us that in Sukhum at the seaside there are a couple of pickpocketers. So overall it is like every other touristic place. Still always felt 100% safe and fine until late at night!
Border from Abkhazia to Georgia
The Abkhaz side here was very friendly and smart talked with us for a long while (15 minutes). But they also asked some questions about our jobs, heritage and other stuff. And they asked us if we did pictures from Abkhazia and/or Georgia. We showed them the camera and he was very impressed of all the places we visited in Abkhazia. But we also had to delete pictures of the border crossing.
The Georgian side just took our passport, checked if we legally came through the same border, too, and then let us wait for 20 minutes discussing with each other. So it was a bit annoying, but not bad at all.
I know many reports like that exist already, but from my own experience I know how much very recent ones are welcome. So I hope this one helps a couple of people, too. If there are any detailed questions just ask me.