You can watch highlights from all Veikkausliiga games on the internet. Iltalehti’s NettiTV is the place to look, usually a few hours after the games (follow ‘muut’ then ‘veikkausliiga’).
Occasionally YLE offers games via their website, I will try to link to them as and when they appear. If you live in Finland, you can see one game a week via the free-to-air sports channel Urheilukanava, usually on Monday or Sunday. Towards the end of the season this switches to YLE1 or YLE2, as they get more interested and start to mess with the kick-off times.
If you live here and want to see Finns playing abroad, there are several options. Canal+ is the home of Premiership football, along with games played at the bigger Serie A clubs and Allsvenskan. Viasat has Danish football, Champions League, Second division English footbal and a fair bit more. Pretty much everything else (La Liga, Bundesliga, lesser lights in Italy) is shown on Urheilukanava.
One excellent programme that may still be available to some in Finland is the TV5 round-up on the French league. It goes out on a Sunday evening, and is in French with no subtitles, but if you did it at school or have watched any football highlights programme before, you’ll get the jist.