Temperature and luminosity
Let's start with some calculations. For the sake of argument, I'll assume that we're talking about a planet that's identical to Earth orbiting a star that's identical to the Sun. To be as generous as possible, I'll assume that this red giant expands to only about $\sim$200 solar radii - a mere 0.93 AU - and cools to around 2800 Kelvin. Therefore, its radius has grown by a factor of 200 and its temperature has dropped to half its value on the main sequence. Now, we can approximate the star as a black body, meaning that it has a luminosity described by
$$L\propto R^2T^4$$
as per the Stefan-Boltzmann law. Therefore, the red giant will have a luminosity $\sim$2500 times that of the Sun - near the lower end of models for our Sun's future. The effective temperature of a planet scales as
$$T_{eff}\propto\sqrt[4]{\frac{L}{r^2}}$$
where $r$ is the distance to the star, and so at its current distance, the equilibrium temperature of the Earth would be about 7 times its current value, about 2029 Kelvin. Now, fortunately for any life-forms on the planet, the radius of the planet's orbit may increase as the star loses mass, which happens significantly faster during the red giant phase. Generously, Earth could move to an orbital radius of $\sim$1.5 AU. If we were to assume an even greater mass-loss rate for this star, it's not too far out of the question that Earth could move as far away as 2 AU. Therefore, if we recalculate the temperature, we find that $T_{eff}$ drops down to 507 Kelvin, or a mere 453$^{\circ}$ F. Hey, it's still better than Venus!
Possible adaptations
I think the classic picture of life on a planet orbiting relatively close to a red giant involves a completely inhospitable surface. I agree; 453$^{\circ}$ F is way too hot for life as we know it to live without shade. Even thermophiles couldn't survive; the famous Strain 121 and Strain 116 - which can live at 250$^{\circ}$ F - would be boiled. I would argue that things won't get much better at night. Heat transfer in a planet's atmosphere is incredibly complicated, so I can't give you hard figures, but the thermal inertia of air would have to be much, much lower than it currently is for any significant cooling to take place at night. Any organisms that tried to live nocturnally would be out of luck.
We therefore have to go underground for shelter, which is a problem, because it means that photosynthesis might not be possible. Chemosynthesis is a possibility, as is thermosynthesis - which, as I've talked about before, was a possible metabolic mechanism for early life on Earth. If I can invoke thermosynthesis here, then I'll point out that, as you'd need a large heat gradient for it to be effective, this planet might be just ripe for it to develop and thrive. As to how exactly it could be implemented - well, I'll leave that to you, for now.
It does seem that cyanobacteria can live deep underground, processing hydrogen gas using a photosynthesis-like chain that doesn't use light. It's pretty incredible, and means that subsurface life is indeed possible. Perhaps these cyanobacteria could survive the extreme temperatures and conditions on the surface.
Periods of rotation and revolution
Now, Kepler's third law states that the length of a planet's orbital period ($P$) is related to its semi-major axis ($a$) and the mass of its parent star ($M_*$):
$$P^2=\frac{a^3}{M_*}$$
Now, we assumed above that $a$ doubles and $M_*$ is approximately cut in half. Therefore, we can see that the new period of revolution is four years.
Now, there shouldn't be a significant change in the period of the planet's day, because that would involve a change in its rotational angular momentum. There's no source for it to offload its angular momentum to, and as its mass and radius will remain the same, so will the length of one day. The one way this could change would be if it has a moon - like Earth does. Over time, tidal forces transfer angular momentum from the planet to the moon, slowing its rotation. You haven't stated whether or not a moon exists; for simplicity, I'll assume one doesn't.
Possible adaptations
I don't see any significant changes happening here. If the length of a day remains the same, then the only difference is that the year has doubled, which doesn't seem terribly absurd. The seasons - such as they are - will also be twice as long, although with arguably no surface life, the change could be tough to see. The only effect this would have on life here would be that the available heat reservoir for thermosynthesis would oscillate over a longer period. That change, though, should be slim, and certainly wouldn't endanger life.
Type of incoming radiation
Dan Clarke mentioned something important, and I'm going to expand upon it. The spectrum of a black body isn't uniform; it's peaked at some characteristic wavelength $\lambda_{\text{max}}$. We can calculated this wavelength from Wien's law:
$$\lambda_{\text{max}}=\frac{b}{T}$$
where $b$ is a constant, 2.897$\times$10$^{-3}$ m$\cdot$K. Using this, we find that the Sun has a peak wavelength of roughly 502 nm. This red giant, with a temperature half that of the Sun, has a peak wavelength of 1004 nm, in the near-infrared section of the electromagnetic spectrum. Moreover, there would be less emission of ultraviolet light than from a Sun-like star.
Possible adaptations
As I wrote about here, different photosynthetic pigments are more effective at different peak wavelengths. For peak emission around 1000 nm, certain bacteriochlorophylls will be efficient. If photosynthesis was possible, the dominant life forms would be purple and green bacteria. Now, it doesn't seem like photosynthesis could happen - the surface temperature is too high. Still, I'm wondering if there's a way around this - perhaps, somehow, the atmosphere is extremely dense and drastically reduces the amount of light reaching the surface to a reasonable amount. If so, purple and green bacteria could be the major denizens.
You asked about mutations. Ultraviolet light is a cause of mutations, and given that the temperature of the star has changed, so has the amount of ultraviolet light - it's gone down. The same goes for any x-ray and gamma-ray radiation (which would be negligible to start with, for a Sun-like star). Therefore, we should see a slight decrease in mutation rates. But then again, there won't be much light out in the open anyway. At any rate, having an ozone layer would be less important.
The AGB phase
The asymptotic giant branch (AGB) phase of a star's life - occuring immediately after the star leaves the red giant branch - presents slightly different hazards for life on an orbiting planet. AGB stars lose mass rapidly through strong stellar winds, sometimes losing up to $10^{-4}M_{\odot}$ per year! This is a major problem. The central stars of planetary nebulae - having just left the asymptotic giant branch - can ablate away the atmospheres of orbiting planets, and I suspect that conditions wouldn't be more hospitable during the AGB phase. In other words, I would conjecture that it's possible that Earth's atmosphere could be stripped.
In addition to the mass-loss problem, there's still the issue of a vastly increased luminosity (perhaps $\sim10^4L_{\odot}$) at the very end of the asymptotic giant branch. Even if the planet retains its atmosphere, and even if the planet moves far enough away to remain in the habitable zone while its parent star is on the red giant branch, it will likely be scorched during the AGB phase.
Possible adaptations
None. Unless that thermosynthesis is working out already.
