From Chisinau, shared taxis (maxi-taxi) and marshrutkas depart frequently from the central bus station for Tiraspol, taking about 1.5 hours. Agree on the price before you get in. At the Transnistrian checkpoint, your passport is checked and you receive a paper entry slip stating your permitted length of stay — typically 24 hours for day-trippers, or up to 10 days on request. Keep the slip with you at all times and present it when leaving.
Most Western passport holders can visit without advance arrangements. Citizens of some countries may need an invitation letter; check current requirements before travelling. The Republic of Moldova does not formally recognise the checkpoint as a border crossing, so officially you are simply moving between regions of Moldova, but Transnistrian officials treat it as an international border.
There is no direct public transport from Ukraine to Tiraspol. The main railway station receives trains from Odessa and a few other points, though schedules are erratic and the route from Chisinau by road is both faster and more reliable.