Getting the right strings for your lute is more complicated that it would be for a guitar. This is because there is a lot more variety in lute building, in the string length and how the lute body responds, so lute strings have to be custom fitted to the instrument. If you go to your local music shop and get a pack of LaBella Historical Lute strings you probably won't be satisfied.
Lutes are strung at roughly half the tension of guitars which means that strings last a long times, maybe years. A guitar player would expect to put new strings on a newly acquired instrument, but if you get a lute, throw away the old strings, and slap on a new set you may be throwing away the result of the former owner's or builder's experience. I would recommend that if you get a used lute you carefully measure the strings, or ask the former owner or maker for a string chart.
Most lute players get their strings from specialists. My "Buying a Lute" FAQ has a list of string dealers, and these dealers can calculate what strings you should get, though you will probably want to use a string calculator like the one at http://www.cs.helsinki.fi/u/wikla/mus/Calcs/wwwscalc.html to help your experimentation.
You will find that many players simply don't use wound strings. I use a gut string coated with carbon fiber (made for harps) that I get from Olav Chris Henriksen. Many people like the gut strings with wire in them that Dan Larson (Gamut Strings) makes. And some people use loaded gut. Mostly people don't like the ringing of the wound strings but the intonation should also be better, though people do complain about intonation of the lower pitched strings a lot. The lower tension probably does not help. Since bass strings last a long long time it is OK to buy expensive strings.
If your action is too high, of course the strings will be out of tune. Lutes need a higher action than steel strung guitars, but it should not be too high - just high enough to not buzz. Unfortunately, lowering the action is usually done by taking the top off and shaving some wood off the sides of the bowl of the lute, so it has to be done by a lute maker. Sometimes you can fix things by pulling up the string at the bridge or pushing it down to get the right height.
The sound of the lute is brighter than the guitar sound and therefore you notice intervals being out of tune more easily. Equal temperament doesn't work as well as it does on a guitar, and you will want to move the frets around until the common chords sound better.