Here are some tips from my "failed" solo climb in early June in 2009 (got to the top of misery hill, saw the summit, and turned back due to AMS). Not sure what your skill level is so I will approach this kind assuming your an average hiker who has done a little research and lives near sea level.
I am assuming your planning on the avalanche gulch route...
Focus on endurance and fitness more than gear and snow skills. If you are a really strong hiker it can be done in one day but I don't recommend it. 11 miles round trip doesn't sound like much but Shasta sure as hell humbled me. Treat it like 15-20 miles. Make sure your boots have at least a 3/4 shank if you use strap on crampons on regular hiking or backpacking boots. My boots had no insulation, a few good socks, but good waterproofing and I was ok. Other wise plan on cold nights and warm days with a good chance of rain and snow even in the summer. Keep it light if backpacking on Shasta since it's a somewhat short yet unrelenting ascent. Sleep at the trail head or near the hut on the approach AT LEAST one night to adjust to altitude. Take short breaks, pace yourself and enjoy the scenery to avoid feeling like shit the whole time.
Basic crampon technique and self-arrest skills are a needed but it's a beginner grade climb. You can refine your skills when you get there. If your a clutz in regular life than walking with spikes on the bottom of your shoes may be challenging. A well balanced athletic person that has never seen snow can do this climb without winter skills starting out. Learn as you go. Your first camp could be in a spot where you can hike up a little bit and practice self arrest skills.
Start early when the snow is firm and keep your cool on the section above the camping area near the lake and below red rocks (10k to 12.5K ish). That is the only really steep part where you would have to self arrest and possibly below the summit.
The two big causes of people getting rescued (or worse) is glissading with crampons still on and rapid weather changes. If your above red rocks somewhere and it goes to white out conditions it would be easy to descend the wrong way onto a glacier or something and trap you up high on the mountain. So bring a map and compass. You can print the map of the summit area from red rocks and up and ignore the rest. Avalanche gulch is a bowl heading straight up to red rocks.
It is also possible to ski down from the summit to the car in the right snow conditions!
Good Luck