First of all, it makes sense to understand whether your final destination is Chop or Uzhgorod? Since passenger rail connection from Cierna nad Tisou (last Slovak station before Ukraine) to Chop isn't that frequent, only twice a day (14:01-15:47 and 23:04-00:50). Travel time is only 46 minutes, but due to time zone difference in schedule it seems like almost 2 hours. Connections from Bratislava to Cierna via Kosice is better and available almost any time of day. Travel time Bratislava-Cierna is app. 8-9 hours.
While if destination is Uzhgorod, then it might makes sense to save time and money and travel cross border by bus instead of train via Cierna-Chop. From Kosice or Michalovce there are frequent buses to Uzhgorod during day time (rail travel from Bratislava to Kosice take from 5 to 7 hours depending on train, to Michalovce - 7-9 hours). Thus such bus option for short cross-border leg will allow you more connections (since in most cases you won't need to wait for either of two trains or arrive 1am to Chop).
Slovak Railways doesn't sell tickets to Ukraine online, but that is not a big deal since anyway all the trains (besides very few InterCity) in Bratislava-Eastern Slovakia connections don't require seat reservation allowing you just to board any train and take any free seat.
Fare and Tickets: here you can have two options:
a) depending on your choice of Chop or Uzhgorod, purchase domestic ticket Bratislava-Kosice/Michalovce/Cierna nad Tisou (respectively 17,18/19,3/19,62 EUR per person). Such domestic tickets can be purchased either at the ticket counter or online from ZSSK website -
www.zssk.sk . Bus ticket Kosice-Uzhgorod cost 6.7 EUR per person. Michalovce-Uzhgorod - 2.6 EUR.
b) in case you plan to travel there and back the same route (meaning Bratislava-Chop-Bratislava), then it might make sense to use such return factor along with your "family of 4". Slovak Railways offers special type of ticket called CityStar for travels to Ukraine, its specifically designed for groups and families, since second, third etc. adult passenger pays only 50% of the first passenger fare (children pay even less, since two children go as one adult). Thus depending on the age of your 4 passengers, it might make sense to use such international CityStar ticket instead of two one-way tickets Bratislava-Kosice/Michalovce/Cierna nad Tisou.
p.s. you probably already have app. timing you want to depart Bratislava (or arrive to Ukraine)? If so, just let us know, then we can already advise you more precisely (with exact train numbers, timetable etc.) Because some of such preferences from your side might as well cancel some options described above