Note that I am assuming you are leaving from Tokyo Station and heading to Takasaki station, since you didn't specify any stations.
@fkraiem has pointed out the cheapest choice is highway bus. My apologies. Taking the Nippon Chuo bus will be 1550 JPY. You have a few stops you can board from, but note that you probably have to buy the ticket ahead of time (there are bus companies you can buy the ticket on the spot, but I don't know about this particular one). Tickets are for only valid for the time you purchases, so make sure you are on time. You can purchase online, through phone, through convenience store kiosks or through licensed sellers. However, this company doesn't seem to have English support, so it may be a task. Information page of that line is here (in Japanese). It will take you about 2 hours, but note that the times fluctuate depending on the traffic and weather conditions.
If you are looking for the simplest solution, it would be just taking the straight route, the JR Takasaki Line (to Takasaki). This will cost you 1944 JPY. If you bought a ticket instead of using SUICA, it's actually 4 JPY cheaper!
Note that this will be a little over 2 hours. Depending on your departure time, there may be a few options with a transfer to get you there faster (by 5-10 minutes).
If you have a JR Pass, you could take the Shinkansen straight there in half the time. The cost otherwise would be at least 4410 JPY. The round trip would be 8820 JPY, so a JR Pass just for this purpose would not be worth it.
There is a Gunma regional free pass (Gunma One Day World Treasure Pass), which you can use to travel freely around Gunma by JR lines. But it costs 2200 JPY, and the closest you get to Tokyo is Fukaya. If will be 1320 JPY to Fukaya from Tokyo, so in total that would be cost you 4840 JPY for the entire trip. Also, the pass is only sold in spring and summer.
In the future, I suggest using a train schedule site to help you plan out, like Navitime or HyperDia.