I only have the Core Set too and it can be difficult to get through even the first scenario with starter decks, but it is possible.
The first thing I did after buying the Core Set was to play Passage Through Mirkwood with each starter deck once. I only won with Leadership and Spirit. I found that Leadership gave me enough allies to fend off the enemies while Spirit kept my Threat low enough and put enough Progress tokens down to get through unscathed. With Tactics and Lore, I was quickly bogged down and overwhelmed as you found.
However, you need a little luck with the draw.
Since then, I have experimented with multi-sphere decks as Rich has suggested. I got through with a mixed Spirit and Tactics deck using Eowyn to quest, Gimli to defend and Legolas to attack. However, this deck didn't do so well in Journey Down the Anduin. I had two attempts and I think setting up lasted longer than the actual game!
For my next deck, I am going to combine Leadership and Lore and see how that goes.
There are a lot of options to explore, even in the Core Set, and I think it's a case of seeing what works for you. There is also a good series of articles called Beorn's Path which are aimed at introducing new players to the game. I got some good advice from there and I would recommend having a look.
I hope you enjoy the game!
Edited by Haarku, 06 August 2013 - 10:09 AM.