After using Brita style filters (slow, expensive filters, bulky) we went with an under-sink 10" filter setup. Similar to what FireHiker proposed with a RO system, but without the expensive RO membrane and about 1/2 the price when you consider the cost of replacing the RO membrane every couple years. The RO systems have the same multi-stage 10" filters plus an additional RO membrane which absolutely everything out, but is very slow (on the order of a gallon or so per hour). To accomodate, most RO systems have a 3 or 5 gallon tank that fills slowly and you draw from.
You can buy single/double/triple setups but if you're on city water anything over 2 is probably overkill (you won't have much sedimentation). FWIW we went with an iSpring 2 stage and we have a 5 micron Carbon BLock filter and a 5 micron graduated sediment filter.
Filters are about $5-8 each and last anywhere from 1,500 to 5,000 gallons (for comparison, a Brita filter lasts about 40 gallons and costs about the same amount). We change filters about 2x/year at a cost of $12. Since its hooked up directly onto my cold-water tap I have water that's filtered to a much higher degree than a Brita on-demand and at roughly 3 gallons/minute flow rate.
If you want to filter your entire house you can get whole-house setups (typically either a set of 10" x 4.5" or 20" x 2.5"). These setups cost a bit more (in the $200-400 range) but can filter 100,000 gallons or more (typically about a year's worth for a family of 4, assuming you don't water your lawn or fill your pool), with each filter costing about $12 or so. The downside is that your water pressure will drop slightly, which may or may not be a concern.
Both under-sink and whole-house setups are pretty easy to install yourself, or any competent plumber can install one in about 30 minutes.
Here's one brand of under-sink filter. If you want 2 or 3 filters just buy multiple and hook them up in parallel.
https://www.amazon.com/Culligan-US-600A-Undersink-Drinking-Filtration/dp/B0002YXMG8/ref=sr_1_1_sspa?ie=UTF8&qid=1535499897&sr=8-1-spons&keywords=under+sink+filter&psc=1