I'd get a 10 inch, decent cross cut (about 60 teeth), thin kerf (thinner blade -- requires less power/wastes less wood -- more suitable for a contractor style saw like the Griz 1022). If you're not doing a lot of ripping in greater than 1 inch stock, it'll be suitable for ripping as well as giving an excellent finish on cross cuts.
Suitable suppliers -- any of the big box places or general hardware stores. Dewalt & Freud have been good blades for me.
In addition to the sharpness of the blade already mention, the burning is also likely to come from binding. Your fence may be misaligned or more likely, the wood may be bowing back into the blade and closing up the saw kerf. This binding and possible kickback, are two good reasons to run a saw with a splitter installed. Splitters are removed for partial depth cuts and then frequently left off.
The 1022 is a good value contractor's saw. The number 0ne addition in my opinion after a decent blade is a dado blade (I'd avoid the wobble type and get a stacked dado).
Have fun and use a push stick.
"If you're doing all the dos, you ain't got time to do the don'ts"