I've been using tangorin when going through core to double check the meanings. I've noticed on numerous occasions that the definitions presented by core are sometimes nonexistent in the tangorin.

I was wondering what J-E dictionary you use or would recommend that would provide alternate definitions. ---Also I plan on making the jump to J-J in the near future and was wondering what your preferred J-J dictionary is? Thanks

I like to use http://www.kotobank.jp when I can't use my EPWING dictionaries for some reason.
http://www.weblio.jp is good too, but kotobank is laid out a little nicer.
My usual J-E dictionary is the 研究社和英大辞典 for words, and 英辞郎(Eijirou) (http://www.alc.co.jp) for phrases.

As for J-J, I have my EPWING viewer software set up to search a bunch at once. 明鏡(Meikyou) is the easiest dictionary I've used (besides the concise one at http://www.sanseido.net, but that one was a little too concise for me), and it includes nice usage notes, but is unfortunately pretty small in comparison to some others.

大辞泉(Daijisen) is one of the first ones I check usually. It's a good general use dictionary. There's another one called 大辞林 (Daijirin) which is almost the same, but I find Daijisen's definitions to usually be a little clearer.

日本国語大辞典 is probably the most in depth, especially if you like word origins, but it isn't as good for more recent words. I tend to use Daijisen more, but Nihon Kokugo Daijiten is great as a backup for words I can't find in other dictionaries.

I actually started writing a guide to Japanese dictionaries here:
http://www.gaiwa.co/2014/06/a-guide-to-japanese.html
But it's not complete yet, so I'm still working on the other parts.

I normally use this one:
http://dictionary.goo.ne.jp/

Non-english, non-chinese learners might want to use this one, which is usually more accurate than the horrible google translate (which first translates to english, then to japanese):
http://translation.sensagent.com/

Personally, I use 英辞郎(Eijirou) (http://www.alc.co.jp) a lot, like Bokusenou. It's got lots of good example sentences.

Generally, I go to ALC first, because most of what I run into these days is a little too specialized, and ALC is great for that kind of stuff. I should go ahead and spring for the monthly subscription at this rate.

It looks like Kotobank and the Yahoo.co.jp dictionaries are combined somehow now? Either way, the dictionaries there are also really good.

I still use my electronic dictionary a lot, though. Having the big Kenkyuusha and the Kojien on it is a huge help, along with a multiple-dictionary search. (Around 20 dictionaries or so.)