2005 Honda Accord Manual transmission problem?

Hi, I have a 2005 Honda Accord 5-speed manual coupe. The car has about 157,000 miles. The car drives fine except sometimes the gears grind. This only happens in two instances. One instance is when the car is rolling very slowly (about 5 mph) and if I try to shift into first, the gears will grind. The only way for me to get into first without grinding in this situation is by pushing it into first quickly and with a slight amount of force or just waiting for the car to come to a complete stop, which isn't always possible. The second instance of the gears grinding is if I'm in high rpms for 2nd gear and I shift into third or if I shift into third too quickly regardless of rpms. The gears will grind unless I wait a second or two between the shift. This only happens going into third gear. My friend who is a mechanic suggested that my synchros were going out and that I might as well buy an entire new transmission. Two other friends of mine who aren't official mechanics said that my car needs a new clutch. I personally agree more with the synchros going out. So:
What do you think the problem is?
What is my best course of action?
Approximately how much will it cost?
Added (1). I forgot to mention this but I always make sure that I push the clutch all the way to the floor.

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