Quick question(s). Where do you live, what's the water table elevation, and what's the frost line? That's important information, as it dictates how thick a gravel base you need to make. The stone's purpose is to prevent and eliminate frost heave, and if it's too thin, you may as well not even have it. W/o knowing where you live, I can't say how thick you need it to be; an easy way is to contact your local city highway department, and find out what their typical detail for a sidewalk is. That's what you need to go by for your patio section to make it last the longest. A lot of cities have their typical details available online. But make sure it's your city, or one nearby, as what's needed will vary due to the 3 questions I asked you first.

As to mixing it yourself vs. a truck... The small quantity you need isn't going to make it worth getting a truck, unfortunately. Don't try to mix it by hand in a wheelbaroow, though; your back will hate you, and you won't be able to do it fast enough, anyhow. Rent a concrete mixer. Once concrete sets up, you CAN'T just add more next to it and call it good. Concrete doesn't work like that.

Since your slab is 5'x10', hand-form (or saw cut down 1", if you're really ambitious) a joint line to make it two 5'x5' panels. It will crack along this line. The goal is not to try to keep it from cracking (that's a physical impossibility); the goal is to prevent it from cracking where you don't want it to crack. Don't put in an expansion joint, here. You don't want to build two completely independent slabs; you want to build 1 slab w/ a transverse joint in it.

While it's not necessary for strength, a wire mesh will help keep the slab together if it does crack. Make sure to keep the wire about 1" below the top of the slab for this to have its effect. Don't let it go any higher than the 1", or lower than the half-way mark. The lower you put this wire, the less ability it will have to contain the cracks. Wire is one of the things people have absolutely no concept as for its proper use; it's placement depends on what it's purpose is. Wire in the bottom of a slab is for strength; wire in the top of a slab is for crack control. Again, this isn't really necessary for the size that you're pouring, but if you do use it, don't put it in the bottom of the slab; it won't do you any good there unless you're building a driveway or something that'll have a hella' lot of point loadings.

Put in an expansion joint around the existing slab where you're butting up against. DON'T pour your new slab directly against the existing; do that, and something going to spall (really nasty surface chipping) and crack, either the existing or your new. I don't know how large your existing slab is, but I'd want around 1/4" to a 1/2" expansion joint based, if it were me. You should be able to pick up expansion joint material at your local Lowe's/Home Depot/etc, but if you can't, then just use a 1/2" thick piece of non-pressure treated wood (you don't want any chemicals f*ing with your concrete curing, aka, chemical reaction). The wood will shrink when it dries out, and provide enough space for the expansion; not to mention, it's soft, and the concrete will squeeze it if it needs to. Do yourself a favor, though; get a good outdoor caulk, and caulk over top of the expansion material (be sure to only use expansion material or wood that is only 3" tall, recessed 1" from the top of the slab) to prevent water from draining down to your subgrade. This will also give it a nice finished look, and will protect the expansion material and make last a lot longer.

Rent a concrete vibrator for when you're placing the concrete. Air bubbles are not your friend, and trying to do it by hand w/ a stick ain't gonna' cut it.