Firstly, make sure you are aware of how dangerous this thing can be. A capacitor bank charged to ~650V can kill so keep fingers well out of the way and make sure it is well enclosed in a non-conductive box.
Secondly, make sure all the components are rated for the voltage and current specified.
Thirdly, there is a danger in that design that after launching the projectile, the current stays ON until you disconnect the AC power source. If it does, the resistors and transformers will get VERY hot and stay that way until they burn out - please fit a switch at the AC input just to be safe.
As for construction:
The transformer connections should be labelled, if you accidentally reverse the output wires of one transformer it will not produce any significant voltage but no harm will be done. If that happens, just swap the secondary wires (right side in the schematic) of ONE of the two transformers.
The diodes will have a distinctive marking on the cathode end. The cathode is the end the 'arrow' in the schematic points toward.
The resistors will be physically quite big. It doesn't matter which way around you wire them, from a functional aspect they work equally well regardless of which end is on the left or right. Do note they may get VERY hot so if possible keep some space around them for air flow. Also try to keep some wire length from the body so your joints don't get so hot.
The capacitors in the schematic are not obvious, the values are not impossible but impractical. I assume they are supposed to say 4700uF but with a voltage rating of > 350V that would be a large and quite expensive component. The schematic shows values of 4.7kF which would make them the size of a small house so I guess they are wrongly marked! Regardless, they will have a positive end and a negative end and you must never reverse them (big bang!). In the schematic, all four capacitors have their positive ends at the top. Others may disagree but I think the 'bridge' over the junction of the first two capacitors should actually be a dot to indicate they are joined.
The launch coil will have two wire ends, if you connect them the wrong way around it will launch backwards! You will need trial and error to work out which way they go. Advice: don't stand behind it on the first trial!
The SCR will have a high voltage across it at all times and will probably have a metal mounting tab on it. Beware that on most SCRs the metal tab is connected to the middle pin so be careful what you screw it down to. To find the pin connections, look for the manufacturers data sheet. Like a diode, the arrow points to the cathode, the side connection is the 'gate' and the other pin is the anode.
The trigger switch doesn't care which way around it is connected but both sides have high voltage on them so make sure it is well insulated.
Other than that, the diagram and construction should be obvious, you just join the components so their pins/wires match the way they are linked in the schematic.
Brian.