You can attach your new colors in any way that works for you. Some sites say to just tie a knot and weave in the ends later. Actually Kaffe Fassett, the 'king of color' says to do that. You can also twist in the yarn for a few stitches before the change and then start knitting with it.
When you need to strand across with intarsia, as, say with a heart that gets bigger and bigger, then you use the yarn you have been and let it strand over to where the new stitches are. You can trap the contrasting color a stitch or two before using it to avoid that smallness of the stitch.
If you consistently have to skip more than one or maybe two stitches in an intarsia pattern, you're better off to have a strand for each section. Often, the space between the two parts that are connected gets bunched.