Welcome to the forum RobertRogers, be prepared for a deluge of information in answer to your question. Everyone will have a different answer for you based on their own experience, you will just need to process the information based on your needs.

If I were to make a list to answer your question this would be it:

First make your quest simple and purchase one of Doug Ritter's Pocket Survival Packs it is a good foundation for a PSK (Personel Survival Kit) for backpacking/hiking. The PSP takes care of a matchless fire starter, signal mirror, whistle and a number of other important items.
Next I would add the following:

1. Adquate water, carry more then you think you need. I also carry something for backup water storage, like Gerber Breast Milk bags. Add some Micropur MP-1 water purification tabA lets and a couple coffee filters to complete your water kit.
2. A full sized compass and good maps of the area your are hiking in. Even if you are planning on travelling well worn trails you should have these items.
3. Headlamp or flashlight, an LED light is prefered with fresh batteries. for Dayhikers I personally think the smaller Petzl Headlamps like the Tikka are best.
4. Locking blade knife or locking blade multi-tool. Doug RItter's RSK MK1 and Leatherman's Wave are my choices.
5. Adventure Medical Kit's (AMK) Heat Sheet and Heat Sheet Bivy are a great addition to the PSP for warmth and shealter. The also double as a signal panel if needed.
6. Dress for sucsess, leave the house with the proper clothing already on and be prepared for the worse weather that could affect you. This includes making sure you have sunscreen, head cover, and sunglasses/eye protection.
7. GOOD footwear, don't skip on quality footwear...
8. A decent first aid kit, AMK makes a number of quality kits for a wide range of needs. I like the Ultra-light 0.7 for day hiking.
9. Your fire starting needs are taken care of in the PSP, but adding a simple BIC lighter or lifeboat matches are a good idea.
10. TELL SOMEONE WHERE YOU ARE GOING. Plan you hike and hike your plan! 90% of SAR is knowing where to start!
11. Expensive but a huge assest to getting found is a PLB... nothing better if you need to be found and rescued!

Hope this is a good start!