Renting a car vs. public transportation

It really depends on two things: how well you can drive in chaotic conditions and where exactly you want to go. Drivers in Georgia are the worst I've ever seen, and Armenians are only slightly better. If you want to go off into the mountains you might need a 4x4 depending on where you go. All the main places in Georgia and Armenia are connected by frequent public transport, with the more out of the way places ranging from a couple per day to a couple per week (i.e. Shatili). Especially in Armenia, there are more options in the AM than the PM. In Georgia, Kakheti and Racha are great places to have a car but personally I wouldn't want to drive in the Caucasus. There are hourly Tbilisi-Yerevan marshrutkas until about 1pm, plus share taxis, plus a daily overnight train. From Batumi you can take a marshrutka to the Turkish border at Sarpi (20mins), cross on foot, get a dolmus to Hopa from where you can easily get to Trabzon. There are also direct buses to Trabzon from Batumi, as well as Tbilisi, but it's said it's faster to go via Batumi and Sarpi yourself and cross on foot as the buses can take ages to get across.

I had been thinking when we fly in to Tbilisi to take a train to Yerevan for a few days, then go back and explore Tbilisi for a few days and then pick up a car. Then go north to Kazbegi, over to Kakheti, then back across to Batumi and Vardzia. Would probably do the public transport route into Turkey and leave the car in Batumi. We wouldn't be going too far off the beaten path.

Do marshrutkis have enough room for luggage?

Are the marshrutkis pretty similar to the ones in Russia?

Am I going to be able to get by pretty well speaking Russian to those who don't speak English?

Marshrutkas are more or less the same in all the the former Soviet states so you'll know what to expect. Some have room for luggage, some really don't. I've seen some in Georgia put luggage on the roof though.

Well, for Kazbegi public transport is very easy, 3hrs and 10 Lari from Didube station. A car would be nice as you could stop somewhere like Ananuri along the way. A car is very useful for Kakheti. Even though all the towns of that region are connected by frequent public transport with Tbilisi, getting around the region is much, much easier with a car. Vardzia is served by one direct marshrutka from Tbilisi and 2 or 3 from Akhaltsikhe but again you would have more flexibility with departure times and could visit places like Sapara Monastery and Khertvisi fortress ruins along the way. So, your route can be accomplished with public transport, however, the quality of the roads won't be a problem and you'll have more flexibility with your own car, as long as you can deal with the crazy driving.

You will be able to get around just fine in Russian. English is not very widely spoken and since I know Russian I don't even bother with English when I've been travelling around Georgia and Armenia.

Came back from Georgia one week ago.
Public transportation is really ok: it is cheap, it's on time, no need to plan (besides train to Batumi in August ;)
Two negative things about marshrutkas: a) they do not stop at nice places where you would like to have a rest, to visit a monastry or to have a snack; b) they drive like crazies!!

NB: No direct connection between Batumi and Vardzia is not possible! I also planned to do it, but you have to start your trip to Vardzia from Tbilisi/Borjomi/Akhaktsikhe.

Speaking Russian is absolutely fine. People of older generation do speak it and very well.