Here is what you'll want:
ID continuous - As high as possible, say around 100 amps or greater. This is the maximum continuous current rating of the mosfet. Select one with a high value to give plenty of buffer room.
VGS TH - Around 2.5 volts or less. This value is when the mosfet will BEGIN to turn on. You want the value to be less than the low end range of your batteries. Being that the range of a single battery/parallel batteries is 4.2-3 volts, 2.5v is an appropriate.
RDS ON - Around 2 milliamps or less. This value is basically the on state resistance of the mosfet. A higher value will cause excessive voltage drop.
As far as looking at the charts, I typically look at one if it is available. The same chart Visus posted which shows the relationship between the RDS ON (on state resistance) and VGS V (voltage supplied to the gate).
Take a look at this mosfet's data sheet:
PSMN1R9-40PLQ NXP Semiconductors | Mouser
Page 6, Figure 7 will show the chart mentioned. Horizontally the chart is showing the voltage to the gate (VGS V), look at the 3-4.2v range. Now look vertically and you can see that the resistance (RDS ON) is around 2 milliamps. You can see below 2.5v, the resistance increases dramatically. That is ok since you should be charging your battery/batteries before or as they approach 3 volts.
Here is a link to the data sheet for the mosfet you tried to use and wasn't suitable:
http://www.vishay.com/docs/91015/sihf510.pdf
ID continuous is 5.6a (too low), VGS TH is 4v (too high), and RDS ON is .54 ohm (540 milliohms! too high).