I want to filter my beer then put it in the bottles for the second fermentation (aka making the beer fizz). I am worried that I will get rid of all the yeast and end up with flat beer.
If you filter out all the yeast then there won't be any yeast to eat your priming sugar and carbonate in the bottle. Though in actuality I would imagine filtering all of the yeast out of your beer is a difficult feat.
If you bottle condition, the yeast in the bottle is going to multiply and create a sediment on the bottom of the bottle. If you are careful with your bottles, you can pour carefully to avoid letting most of this sediment into your glass.
But if you want to have absolutely no yeast in your bottle then you are going to have to find some other way of carbonating your beer (eg: force carbing via keg).
If you are a homebrewer, I would recommend avoiding any sort of filtration when bottling or kegging until you're a very advanced brewer. You will need to force carbonate if you filter.
If you are simply trying to get rid of some of the cloudiness, make sure you use a yeast that is not low flocculation (medium is ideal, because highly flocculant yeast might need to be roused if conditions aren't good).
You can indeed transfer your beer to a secondary fermentation vessel (i.e. another carboy) and age it for a while for additional clarity. You can also leave your beer to settle in the bottling bucket for 30-90 minutes after transferring from your fermentor to allow everything to settle.
Ultimately, I rarely worry about clarity in my beer. If it tastes good, it's a successful beer. That said, I usually end up with clear beer because I do most of what I suggested, though I tend not to rack to secondary anymore and instead let my primary fermentation go for 3-4 weeks (you're not going to get any off flavors from yeast in that time, but it will clear up some).
Since you're primarily looking for clarity in your beer, then the easiest solution is to cold-crash it.
Cold-crashing basically consists of:
The cold temp will really help clarify the beer. It doesn't harm the beer or affect the flavor at all, and there's still plenty of healthy yeast in the beer to handle bottle conditioning.