Do NOT go via Tiraspol...
DO NOT CROSS FROM MOLDOVA TO UKRAINE/ODESSA VIA TRANSDNIESTRIA - the autonomous republic that now separates Moldova from Ukraine near Odessa. I have done it but STRONGLY DO NOT RECOMMEND YOU TRY.
Visa restrictions to Moldova for EU citizens were lifted in Jan 2008 - you can now visit for 30 days without applying for a visa in advance. But, because of Transdniestria, there are now NO train services from Kishinev to Odessa (there used to be many) and the crossing must be by bus (or car). An alternative, far safer route, is to travel down the south of Moldova to Ukraine, near Izmail, then up to Odessa - it takes the same amount of time (about 6-7 hours city to city). You can also cross into Moldova safely from the West of Ukraine (via near Chernivtsi).
If you are travelling from Odessa to Lvov, don't worry about being taken off the train in Transd. in the middle of the night - which used to be the case. All train routes have now been redirected to avoid Transd.
If travelling in the South, check which route your bus takes if you want to travel Moldova/South Ukraine: if it is via Tiraspol - forget it for your own safety. Because Moldova does not recognise Transd. they will not issue you with a visa for entering the country. This will cause problems if you enter Moldova via Transd. then want to leave via a different border.
Foreigners are NOT allowed to visit Transd. The information you will receive at the border will differ depending on who gives it to you and the mood they're in, but, technically, as a foreigner (if you're lucky enough to get hold of a 'visa' to transd - only available at the border - you will only get a transit permit (a slip of paper), valid for 2 hours - about the time it takes the bus to pass through).
Note: IMPORTANT: People go missing in Transdniestria (and having passed through, I can understand how and why)... You WILL have to bribe your way across. OR you WILL be turned back from the border - just refused entry because you are a foreigner, and there will be nothing you can do unless you want to look down the barrel of a gun. You may be imprisoned or have your passport and possessions taken by border control. (Please note: Moldovan and Ukrainian border guards are just as bad as their Trandniestrian counterparts at this border: other borders they are ok). There are no foreign consulates in Transd., no embassies, no one you can turn to for help because no other countries recognise this place as existing. But... it does exist and it is amazing to think that this kind of lawlessness exists, these days, on the doorstep of Europe and yet no one does anything about it.... If you're thinking of heading by car through Bulgarian borders (especially in the South) you can expect the same kind of treatment...
Currency: the Moldovan Ley is only available in Moldova. You cannot change it anywhere else. (If you're lucky one exchange counter beside the bus station you get off at in Ukraine will change them, at a not very favourable rate). It's best to get rid of them in Moldova. There are plenty of ATMs in Kishenev if you want to take Ley directly out of your home bank account.