I have a Telecaster Standard which I bought in mid-December and changed the strings on Christmas day from the original Fender strings to Ernie Ball regular. I've been playing it more recently and noticed that it sounds out of tune if I put a capo (Kyser brand) on any of the first 4 frets. I made sure it was in tune before attaching the capo and after as well but it made little difference with the capo.

A separate problem I am having is with my tuner. When I plug my Monster instrument cable into the Korg tuner, the needle moves to Bb# and continues to stay there until I hit a string. I've tried turning the volume knobs down and moving the selector into different positions but there is no change. This is more of an annoyance because if I hit E it'll be right in the middle but if i do it a few seconds later it will be a bit sharp or flat or it will move around randomly from left to right. The guitar sounds in tune but the tuner is never static. I've never had a problem with this tuner before with my acoustic so this is new to me.

Can anyone help?
what i would do is find a good guitar tech in your area and have him do a complete set up,but before that make sure your new strings are locked in good on the tuner post on the head stock, the guitar wont stay in tune if there not.
I restrung my guitar yesterday and it stays in tune much better now than it did before. However, if I put a capo on say the 4th fret and play F (A without the capo) then it is displayed as almost completely sharp on my tuner. So basically every note is sharp when I use the capo. Could this be because my capo is not pressing the strings down tight enough?
your capo must press down so that the stings will be able to sound a note with out buzz or being muted. you can try tuneing down a haft step then capo the neck.i have jumbo frets on my strat and it sounds sharp when i press down to hard so i tune it to e flat and it sounds alot better.
maybe an intonattion problem

try to put you capo on the side of the fret that's closer to the headstock