Walk on the Zuruldi massif

From Zugdidi to Mestia:
Marshrutkas to Mestia leave from Zugdidi every day, there is at least one in late morning and one in the afternoon. Morning marshrutka departs from the bus station situated by the bridge around 10 am (or even later if not full). Afternoon marshrutka waits for passengers of morning train from Tbilisi, which arrives around 14:30. Therefore, they wait in front of the Zugdidi train station. There might be some odd marshrutka in-between on you might get lucky and board marshrutka stopping at Zugdidi along its route to Mestia, but that cannot be guaranteed.
Either way, the price of the ride is 40 GEL and the trip takes almost 4 hours, mostly due to a rather large pause these buses tend to make around Khaishi.
If you get lucky or travel in a larger group, it could make more sense to secure your spot at shared taxi that depart from the station by the bridge (and in the afternoon from train station as well). The price tends to be 200 GEL per the whole car (50 GEL pp if the taxi is full), but varies depending on the situation. Trip is faster since any break is optional, takes 3 hours or so.

From Tbilisi to Mestia:
Few years ago, the best option used to be the overnight train to Zugdidi. However, it was cancelled in 2020 due to covid and there are no signs that it's going to be restored anytime soon. The only viable replacement used to be a long-distance night bus, which could also save you a lots of time, but won't be as comfortable as that cancelled sleeper train. Unfortunately, it seems that right now (summer 2023), they are not running them either, only bus leaving around noon, but it might be worth checking out Omnibus website to see if they were reinstated.
Currently, I would recommend the day train. It departs from Tbilisi at 8:40 and makes it to Zugdidi at 14:34, just in time to catch some of the last marshrutkas to Svaneti. The price of the ride starts at 16 GEL, which is less than 5 EUR. Tickets can be bought in advance at matarebeli.ge website (the official web is railway.ge, but that never worked for me). Just one important thing - tickets are released for the online sale only some 15 days in advance, but it's not explained anywhere. So don´t freak out if you can´t buy your ticket online because of the "no tickets available", "trains not found" or some similar error. Just try later :)
If you fail to secure the spot on the train, you may be forced to take a direct marshrutka - two minibuses depart at 7:00 from the area in front of Main Railway Station, come at least 30 minutes earlier to catch a spot. Another one departs at 8:00 from Samgori railway station and there should be something from Didube as well. The price is 40 GEL, so it´s quite cheap, but I can´t recommend it since this uncomfortable journey takes grueling 9-10 hours - basically, it's just slower, marginally cheaper and much less comfortable alternative for the day train.
If the are not travelling alone and don't mind spending some extra bucks, you might prefer a direct taxi. Those can be either arranged at bus stations or prebooked - looking at the prices of one of the most popular Georgian transport website, gotrip.ge, prices for the private transfer from Tbilisi to Mestia start at 650 GEL (230 EUR) for the group of 4.
The last option you could consider is flight - there are 4 flights a week from Natakhtari airport near Tbilisi, operated by VanillaSky company . You would see Caucasus from a completely different perspective and for 90 GEL it´s quite a bargain. On the downside, they occassionally get canceled because of the weather and during the summer, you should book at least month in advance since they get booked up pretty fast. For more info, pls check this excellent post about the flight (covers also flight from Kutaisi).

From Kutaisi to Mestia:
This depends on whether you want to get to Mestia from the city or from the Kutaisi airport.
In the first case, your best options is the direct minibus (the price is 40 GEL) departing from the station behind McDonalds. There is a fixed departure between 9:00 and 10:00 am, but if you want to secure your spot, better get there at 8:00-8:30 since it's popular among tourists and will depart much sooner if full.
According to locals, if this marshrutka gets full and leaves before 9:00 am, they will send another one. If it gets full later, they might not.
There may be also others during the day if there are enough tourists, but I wouldn´t rely on that. If you miss them and there is none other direct bus available, look for a marshrutka to Zugdidi (price 10 GEL) - it's the same way and you can find there transport to Mestia more easily (especially if you notify your driver in Kutaisi that you need to continue to Mestia).
If you are travelling directly from the airport, you also have some options. If you are travelling alone, you can arrange a transfer to Kutaisi at Georgian bus stand (5 GEL or so) and take marshrutka from there. Or, if you are willing to spend extra money and save time, find someone to share taxi with. It's actually pretty easy - do you see any people with hiking backpacks? If yes, there is a pretty high chance they are heading to Svaneti so this is a great chance to save some money and get to know some people.
If you are not alone and travel straight from the airport, you might want to book your taxi in advance. In this case, at gotrip.ge, prices of the transfer from the Kutaisi airport to Mestia start at 340 GEL, which is actually pretty solid. If you haggle with taxi drivers loitering around the airport and don't look to desperate, you might get a better price, but not by much.

From Batumi to Mestia:
Marshrutka is the only option, it departs from the main bus station. Since you will have switch the bus in Zugdidi (you may have to wait for several hours there), this ride takes a better part of the day. The first bus departs at 8:00 am, then there should be another one at 9:30 am.

The first marshrutkas to Zugdidi, Batumi and Tbilisi depart early in morning, around 7 am. every day in the morning at (these should be booked one day in advance). There are more during the day, depending on the demand. There should be also some marshrutkas departing in the afternoon to Zugdidi, just in time to catch the evening train to Tbilisi which departs at 17:25. However, since marshrutka schedules are not 100% fixed, better visit station in advance to confirm this and book tickets.
For exact timetables and booking of the tickets, visit the travel offices at the station. Just keep in mind that there are several offices and don't buy tickets too far in advance. We met tourists who entered the first office, bought there tickets for the next bus which departed in 2 hours and then had to wait even though there were several other marshrutkas (operated by another offices). If you can't find a suitable tickets and are not under a time pressure, I recommend just to grab a beer at the bar by the station, sit on the terrace and check the situation once in a while - sooner or later, some driver driver will show up.
If you prefer to have everything pre-planned, you might book a taxi in advance, especiallly if you are not travelling alone. The cost of prebookeded private transfer from Mestia to Tbilisi starts at 550 GEL. Of course, you might get an even better price if haggling with local drivers, it all depends on who you stumble upon. Svan drivers are often quite reckless - if you don't like the driving style of your driver, don't be shy and say so.