Well, we have buses, trolley buses, coaches and taxis within the city.
COACH ( Russian word for it - 'marshrutka' ) is a mini bus which can hold up to 15 people. Last time I was in Hrodna the price was 850 rubles per person (~$0.4). You pay to the driver upon entering or you can pass the money through people after you sit down.
Don`t be surprised to be tapped on the shoulder by someone from behind - they just want you to pass the money to the driver.
Don`t sit on the front seat, you`ll get tired of passing those rubles every time new passengers come in. ;-)
It does not matter if you are a child or an adult or if you go 1 or 10 stops - the price is the same for everybody. Of course, toddlers may sit on their mama`s lap for free.
Trolley buses and buses are usually crowded and it is not so rare when you can smell someone`s sweaty underarm next to you or a bad breath or worse - being harassed by some drunk ass..s.
I personally prefer to take coaches.
BUSES and TROLLEY BUSES:
You can enter trolley buses and buses without having to show a ticket ( 'talonchik' ) but if controllers walk in, you`ll be in trouble. They are hungry for your cash! Remember to put talonchik into the small device hanging on the bus walls to punch it. Before that exercise your talonchik is not valid.
If you refuse to pay, you`ll be asked to get the hell out of the transport and put into a small bus with the rest of the violaters until you pay the fine.
So, just buy 'talonchik' (500 rubles (~0.23) at the nearest kiosk and relax. Kiosks can be found not on every bus stop but you can also purchase a 'talonchik' from a driver. Too bad that very often the driver does not have it. Oh, well.. life is tough.
If you are from the West, it`ll be affordable for you. A 15-20 minutes city ride will cost you about 10,000 rubles ($5) during the day.
Minsk Central Bus Station is right next to the Railway Station (Bobryjskaya Str., 12). Look at the picture.
Buses from Minsk to Hrodna start running from Central Bus Station at 6.10 am (every day) up till 8.30 pm (every day). There is basically a bus/coach going to Hrodna every 30-40 minutes or so.
You have two choices: to take a slower bus (5.30 hour ride) or a privately run minibus (3.30 - 4.00 hour ride).
Minibus will make only one stop, somewhere in the middle, so the passengers can go to the 'bathroom'.
It`s the priciest (after private car) option - 36,000 rubles one way (~$17) vs 18,000 rubles on a train.
NOTE: Since the Cenral Station is closed for reconstruction, buses now leave from Moskovskaya Bus Station (to get there take a subway and get off at Moskovskaya Station).
There are about 5 different trains that go from Minsk Central Railway Station (on the picture) to Hrodna.
The fastest is the daily 'Minsk-Hrodna' train #127. It`ll bring you to Hrodna within 5.40 hours. Departure is at 4.22 pm, arrival to Hrodna at 10.02 pm. In April 2008 its cost was 18,000 rubles (~$8.5) one way.
If you need to arrive to Hrodna in the morning, then you`ll have to take a night train. The earliest you can get there is at 5.13 am on the 'Minsk-Hrodna' train #615 which leaves Minsk at 9.17 pm. This trip is 7.56 hours long (the longest actually).
You can arrive to Hrodna a little bit later (at 6.11 am) via 'Unecha-Hrodna' or 'Vitebsk-Hrodna' traines #391 and #683 accordingly. They both leave at 11.20 pm and it takes them 7 hours to get you to Hrodna. The only difference is that 'Unecha' runs on even (2,4,6..) dates and 'Vitebsk' on odd (1,3,5...) ones.
There is one more train - 'Moscow-Hrodna' train #77. It`ll get you to Hrodna at 9.37 am BUT you`ll have to get on it at 3.05 in the morning. This one rides for 6.30 hours.
You can take a bus from Hrodna to: Byalostock (Poland); Warsaw (Poland); Lodz (Poland); Druskeniki (Lithuania); Ivanovo (Russia, via Moscow); Kaliningrad (Russia); Vilnus (Lithuania); Kaunas (Lithuania); Prague (Czech Republic).
For us the best way to go to Grodno was by train. My girlfriend live in Poland and we were travelling in the east of Poland anyway. From Bialystok you can take a train in the morning around 10:00, you have to exchange 2 times (in Sokolka and Kuznica Bialostocka), but you arrive at a decent time (13:40).
Later there goes a direct train (14:50), but you arrive much later too (at 18:30), which is maybe not the best idea when you don't go in the summer and it already is dark in the evening.
When you arrive in the train station you are being guided to a hall with custom control, where you have to put your luggage on a table (but in our case they hardly checked it). I only had to wait a bit to get my passport back, because they had to add a paper with Russian translation of my data. Later I had to buy a special health insurance in the kiosk near the station (Belgostrada). After that we took a taxi to our hotel.
Hrodna's bus station and railway station are quite close to each other and it's probably where you are supposed to arrive unless you drive on your own. The bus station is under renovation and is composed by two buildings, one for local routes and one for long distance and international routes. So you have to go to the right ticket office in order to get your tickets.
Understanding the timetable is never easy in forme Soviet countries, and Hrodna is not an exception.
There are several international connections from Hrodna, mainly to Byalistok, che closest bigger town on the Polish side. There are connections to Warszawa, Vilnius, Kaunas, Kaliningrad, Moskow, Praha and Riga too.
If you can read in cyrillic and understand (it's really easy) the names in Belarusian language, you can find a very useful timetable HERE
For national routes it's usually better to take a marshrutka, there's plenty of them, expecially connecting Hrodna to Minsk and Brest and all the smaller towns nearby.
Talking about trains, the international routs have been almost all suppressed, while there are some national routes, mainly in the direction of the capital.
The best way to discover Hrodna is surely on foot, and the town is really not big so it's not so tiring to just walk.
But if you are tired there's plenty of trolley buses and marschrutkas which will take you basically everywhere. The trolleys are really old and in bad conditions, slow, but they still work. Marschrutkas are usually much faster, but it's not easier to understand where they go, expecially if you don't know Russian/Belarusian.
The train and bus stations are not so close to the centre, so taking a bus could be a good idea, expecially if you have luggage. The Hotel Belarus is even farther away and up on the hill, on the opposite side of the station. By the way I walked from the Hotel to the station on my last day there and it didn't take more than half an hour, with my big backpack (ok, I walk fastly, I admit it).
The busses are old and dirty and sometimes very full, but it is really the best way to travel around Grodno, especially because the hotels are not that nearby the center. It is a boring road to walk all the way and more than 2 km., so taking a bus saves a lot of time. The busses are so incredibly cheap, that you don't have to worry about the money, a single ticket costs 220 Belarussian Rubles, which is about 8 eurocents (0.08 Euro).