I can share a couple of free tools I find useful (other than Google Keyword Planner)...

Google Trends is always a good broad tool to look at to see where popularity for those keywords are at. Of course your keywords will need to have enough volume, but you can get pretty good segmentation now in GTrends.

Ubersuggest is a great tool to find a huge list of related variations to your keywords. Very useful for planning out content.

Moz's Keyword Analysis tool I find is very useful for judging the difficulty of the keywords you are researching. You want to target keywords that have enough traffic, but also are not to competitive if you are starting some initial optimization.

SEMRush is another useful and popular tool for keyword research. You can get traffic estimates and competition level estimates for a large index of keywords in localized Google serps. It is free for limited data - paid for full results.

You can also use tools like Fresh Web Explorer, AllTop, Topsy and even Google Alerts to give you ideas of popularity of keywords and different content groups.

TL;DR Moz is great for difficulty and competition research, but there are better tools (and free!) on the market for traffic and popularity measuring - use a combination of both!

Hope this gives you some ideas!

Your header images should not have h1 tags in them as well because then every page is going to have the same h1 tag with the same keyword - making it so that your are optimizing against that keyword on all your other pages.

Google looks at h1 tags as clues to what the page's content is about. If you have multiple h1 tags with different keywords then it is difficult for Google to contextualize the page.

Best practice: one h1 tag with the keyword or theme you are trying to optimize for.