Definitely replace the cables, and I'd go for new outers to keep things fresh, considering the issues you've been having. Index them properly, checking your limit screws are actually set right.
When installing new cables I make sure the shifters are in the smallest gear, and pull the cable as tight as possible, clamp it into the mech, then shift up and down the gears to pull out the cable stretch a bit, then shift into the smallest gear again and take up the slack with the barrel adjuster on the shifter - is the cable is just tight in the smallest gear, then the rest should index perfectly.
This assumes you've set the limit screws correctly.
If you can shift fine now, but the chain is slipping - check the chain wear as below:
Grab a chain stretch tool from somewhere if you can, assuming you don't have one: measure your chain with a ruler (12 linked pairs should measure 12 inches from new 304.8mm, if your chain measures over 1.5mm extra on top of this at 12 links, then you need a new chain, if it's 2mm or more stretched, then you need new chain rings and cassette too).
If you don't have any slipping issues, but the chain is stretched under 2mm, then replace the chain only. If the chain is stretched over 2mm and everything else is fine, then just run the drivetrain into the ground, it'll last a good length of time and save you a shed load of cash.
Good luck buddy.
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