Shimano road shifters have to be matched with road front derailleurs, and mountain shifters have to be matched with mountain front derailleurs (technically, you could use a pulley like a JTek Shiftmate, but thats a dumb solution due to price reasons). Theres no compatibility difference in the rear (update: see the comment - the following holds up to 9 speed between mountain and road), except mountain derailleurs are normally long cage and can take bigger cassettes (and thus are favorable in a touring context). Stick with one brand for shifters, front derailleur, rear derailleur (crossing brands is possible, but more complicated as there is no standard for how much cable shifters have to pull, and you may need pulleys to change the cable pull to get things to work).
Front derailleurs are normally also matched to if they are double and triple and are specified for what range of chainwheels they work most favorably with (you can find this on the data sheet for the front derailleur - this is less important with friction on the front, but brifters can be finicky). In choosing a front derailleur, you also should make sure that the front derailleur can be mounted on your frame and the cable pull is in the right direction for your frame.
The particular crankset you pick's compatibility will be determined by if you have a bottom bracket that matches it in the bike, along with if its a double/triple (matching the front derailleur) and within the appropriate tooth range and has appropriately sized tooth jumps. To quote Sheldon Brown (article linked later): "Shimano's "10-speed" triple front derailers are optimized for a 13 tooth difference, typically 52-39. Most other Shimano front triple derailers are optimized for a 10 tooth difference between middle and large ring.". The speeds on the front derailleur are marketing term - you don't really need to stick to them since the speeds are determined by whats in the back(but obviously, Shimano wants you sell you an entire groupset, and 10 speed systems use narrow chains, so you should have a 10 speed FD, 10 speed RD, 10 speed chain (you could use a lower speed FD, but it might not be as crisp) ).
I recommend reading this article for seeing what goes into selecting a front derailleur and this one as well.
In terms of price, the shifters will be the biggest expense and will determine the FD. Regarding the RD see the comments, so you don't need to worry about it. Once you pick the shifters, you can pick a relevant FD and crankset.
Also note that a 2x10 or 3x10 setup is pricey relative to many 2/3x8 or 2/3x9 systems- the chains don't come with master links by default (and are pricey), and since things are narrower, they can be more annoying. Will the extra 1 in the rear buy you anything? Probably not.