Targeting to low competition keywords is a perfect idea. Tons of this traffic will give you more conversions than highly search keywords. In addition, you will spend a lot less time to promote this type of keywords.
There is no OR here. A good keyword research would involve seeking highly searched AND low competition keywords. The combination between the two is what counts. Low competition can give you a ton of keywords you are not interested in, after all relevance is what matters right? High Search will give you good keywords but some you are close to impossible to compete with. Get the idea...?
Low competition keywords are the first choice. Then between those keywords should be chosen the ones most searched. You'll have the idea to focus on important keywords, but these are very difficult to compete with. You can try anyway, but it will be very complex to rank.
I would start with the "low hanging fruit"/low competition keywords first, plus the low comp longtail keywords. Depending on your site's Authority and amount of competition, you can rank first page with zero or very few backlinks. Then of course go for the highly searched keywords after you've targeted all the low comp ones. Normally you'll get more traffic from low comp keywords than you will from high comp because of the better rankings.