I'm trying a new tactical character, and I'm trying to progress through the game on Elite difficulty.

I've done a few missions, and I cannot seem to beat "Stop the Signal" on elite. I just don't have the firepower to take down the Undine dreadnaught at the end.

So, I decided I was too low level, and went to explore the Delta Volanis Cluster. I had a ground mission there and I simply could not defeat the last group of klingons. Not with the crew that I currently have. My crew simply dies way to fast.

If this were end-game I'd say it's my gear or strategy - but I'm Lt. 5 and there simply are not a lot of options for my crew at this time.

So - am I missing some strategy here, or is Elite really supposed to be this hard? Don't take me the wrong way - I don't mind a challenge - but this seems impossible at this point.

With Season 5 patch, the mission became a tad harder. Ally yourself with a few other players, or one high level player, and your problem will go away... litteraly.

Right, but that defeats the purpose of me trying to defeat missions on elite.

They boosted the damage resistance and damage output of enemies for Elite Missions. If you want to beat them, you'll either have be team up with other people like Anazonda (a team of 3+ should be enough), or have blue/purple gear. For a singular player, it's extremely difficult-if not impossible in some cases- to deal enough damage to take out explore "bosses" or the Dreadnaught in "Stop the Signal".

So, advanced difficulty would be a better option for solo play?

Depends on the playstyle and level of 'challenge' you personally find acceptable - and there's no hard and dfast/right or wrong answer, it's all personal peference.

Me? I still find Elite too easy the majority of the time and wish they would take a cue from Champions Online and add a 4th, even HARDER difficulty level to STO.

Try Advanced in an explore and see if its to your liking. It won't be too hard, provided you have equipment and fairly decent weapons. At your level, something beyond the Standard Issue items will suffice. It'll be less likely to drop uncommon or better lot than Elite, but better than Normal.