The direct answer is that you do not need a resistance in series when powering a 5 volt relay coil from 5 volt power supply.
-- For more in depth understanding, go to the data sheet that you linked, and look at the coil data. -- From the coil data, you can calculate the amount of current required for the coil.
---- Lots of in depth analysis can be made depending upon how you drive the relay coil.
The relays on the shield you linked are rated for 7A at 250V AC, but I don't think the relay shield itself is designed for that.

The product page states:
Cautions:
Place 2 layers of electrical tape on the top of the Arduino's USB connector. This will prevent the relay shield from making contact. Do not operate voltage more than 35V DC.
(Emphasis mine.)
The datasheet for the relays on that board (Heli Shun HLS8L-DC5V) show that the primary coil is 5V 0.36W:

Note that the contact ratings section specifies 250V AC / 30V DC. This immediately makes me question the difference between the relay datasheet (30V DC) and the shield product page (35V DC).
Since you're using AC, this discrepancy might just be a side note. The shield product page doesn't even mention AC for the switching voltage, it just repeats the 35V DC max switching voltage.

Based on this, I don't think I would connect 240V AC, because it may be that the PCB isn't designed with the necessary voltage isolation.
Further, since you're not experienced working with mains voltage, which can be extremely dangerous, I don't recommend doing so without the assistance of a professional.
Best Answer
Add a bridge rectifier and it should work OK.
simulate this circuit – Schematic created using CircuitLab
You can use four diodes or a bridge rectifier which may be neater.
[Update after comments]
Capacitor is not required. Provided no smoothing capacitor is added the rectified 24 V AC will be have wave rectified 24 V RMS (ignoring diode losses). The heating effect in the coil of 24 V RMS is exactly the same as 24 V DC so no problem there.
Addition of a capacitor will result in slow drop-out and slow opening of the contacts. If the load is significant relative to the contact rating then arcing and premature failure will occur.
Addition of a capacitor will also result in higher DC voltage - approaching DC peak - and this will overheat the coil. Half-wave rectification requires addition of a capacitor which is a bad idea as discussed above.
Relay chatter isn't a problem because
Reverse diodes to protect the bridge rectifier aren't required as the function is provided by the bridge itself.