The first thing to determine is who will be using the basement and what the primary use of the basement will be. Spend some time on this considering not only the present but the future as the family grows, or grows up.
Whenever possible do not use a suspended ceiling as this gives the feel of an area separate from the rest of the home. In newer homes, 25 years old or less, items like plumbing, electrical and HVAC should be up to date from a code standpoint and new enough that it will not need repairs for some time. In homes older than this a suspended ceiling may be a better idea as repairs to existing XXXX may be required sooner than later.
Most basements are darker than other parts of the home due to the lack of adequate windows. Help this by using lighter color paints and adding more lighting than would normally be considered in the ground level rooms.
It may help with head room if some of the plumbing and HVAC ducting is relocated to the side walls where it can be hidden with soffits under which wall cabinets or shelving can be built.
In today’s basement build outs a bathroom is a must. Ideally build a bath with a shower even if you are not adding a bedroom. The next owner may want to change part of the basement into a bedroom.
Creating an open stairwell without a door at either the top or bottom will make the basement become more a part of the existing home.
Use a separate heating system if possible with it’s own thermostat for controlling the temperature in the basement.
Basements should have at least two exit area. In older homes it will be necessary to cutout one of the existing windows and install an egress window with a built in ladder for ease in exiting. Each bedroom must have an egress window or other exit.
Flooring is always an issue.. Carpet is the least expensive, but is the basement has any water issues or plumbing from above leaks you may have to replace the carpet. Wood flooring is usually a no-no as replacing it if damaged is quite costly. A composite floor or tile floor is much more expensive, but will probably never have to be replaced.