Your main thing would be knowing how much current in total your project will draw; once you know that, then you can decide how long you want it to run on battery, in order to figure out what the total amp-hour rating of the battery should be.
For instance, that battery is an .8 Amp-Hour (800 mAh) battery; what this means is that if your circuit draws (on average) 800 mA, you will get 1 hour of run-time (this is simplified - more on this later). If your project only draws 400 mA, you would get 2 hours of run time. If it draw 1600 mA, you would get .5 hours of run time. See how that works?
Now - the reality of the situation is different. You will -never- get that perfect run time; due to internal resistance of the battery, and other factors (external temperature and the heating up of the battery as it is used, among others), the actual run-time will be much different. You also should adhere to the C10 rule of charge/discharge - otherwise the battery could be over-stressed and ruined. The best way to figure out all of this, would be to consult the datasheet for the battery; it will have all the information you need to determine whether it will fit your application (of course, you -must- know how much current your project will draw before you can do anything - so figure that out first).
Now - as far as charging a lead-acid battery? Once again, you need to follow the C10 rule for the battery; if you don't you can damage the battery. If you can follow it, then you can plug in an appropriate 12 VDC wall-wart capable of supplying the C10 current needed (or less - you never want to go greater, unless the battery is rated for it, or you don't mind shortening its life - this is a so-called "quick-charge" mode, some batteries can handle it, some can't - but in just about all cases, it wears the battery out quicker) - and then charge it for the needed time (but no longer - you can just plug it in and let it go).
This only applies to lead-acid batteries, by the way - if you want to use NiCd or NiMh (or if you want a "plug it in and forget it" charging capability with lead-acid), you would need to build a charger. If you want to use a rechargeable lithium battery of some time (Li-ION, Li-Poly) - those need special chargers, and are really outside the scope of a beginner (because if it isn't done right - such batteries have a nasty habit of explosion and fire).