It depends on the circumstances - after all if one cable was best for all situations then that is all that would be on sale.
Solid core is cheaper than stranded and less vulnerable to corrosion, not that that is usually a problem. However it is also considerably less flexible. Worse, if you _do_ repeatedly flex it it is susceptible to metal fatigue and eventually cable failure. Stranded cable avoids those problems. Never use solid core cable for patch leads or for fixed wiring where the cable emerges directly from the wall for the final run to the networked device. For fixed wiring to wall mounted face plates you can use either kind subject to using the correct face plates - the punch down blocks on some face plates are only suitable for solid core cable and will give an unreliable connection with stranded cable, so be sure to check up on this before installation.
As for screening it really depends on the circumstances. For today's networks UTP will generally be fine unless the cables are being routed near electrical machinery - anything with large motors - in a domestic setting that usually means fridges, freezers, washing machines and tumble dryers, although even those are generally small enough not to cause significant problems.
However, it really doesn't make sense to install cat6 without an eye on 10GbE, since even if you are not using it now that must surely be the intention in the future, since otherwise you can happily use cat5e. 10GbE over cat6 has slightly limited run lengths compared to today's networking standards and that gets worse in "hostile" environments which can simply mean many cables in the same piece of trunking. Screened cable - FTP or STP - will improve the distance you can get in such environments so if you have runs longer than perhaps 40-50 metres it is worth considering. However, it is more expensive and thicker than UTP cable (meaning larger trunking for the same number of wires) so I wouldn't use it without good reason.