Did you make sure you replaced the strings with ones of the same gauge? I ask because it sounds like you may have put on a heavier gauge which has caused the neck to bow with the increased tension. It's not a massive issue but if it is the case you will need to adjust the truss rod. It's a bit of a daunting process but honestly it's not too hard, there's plenty of guides out there on how to do it. Or you could take it to a shop if you're not feeling too confident.
(Edit: after reading your comment again it's more likely you replaced with a lighter gauge if the strings are hitting the first couple of frets, same solution though just turn the truss rod the other direction)