arimaautocorrelationtime series
Can you please tell me what should be the values of ACF and PACF from the graphs I have attached? I think it should be ($p=0$, $d=1$, $q=3$). I have differenced the data once so $d=1$ and there are three spikes in ACF so $q=3$. But function auto.arima from "forecast" package in R is giving the answer as (0,0,1).
Also, what should be the values for the seasonal $P$, $D$, $Q$ and its period value?
The short answer is to fit an AR(1) model & check it. If what you're left with after that is pretty much white noise, you might well be safe to assume they're AR(1) - if that's a reasonable model a priori, & depending on what it is you're wanting to do with them.
The ACF & PACF suggest, however, that there's perhaps more structure there than a simple AR(1) model. You shouldn't necessarily be bothered about the fourth lag in the PACF being just over the 5% significance level (assuming that's what the blue line is - you didn't say) - there's no correction for multiple testing, so in 20-odd lags you'd expect that. But the wavy ACF could indicate you need either to difference or to put in at least an extra AR term. Given how slowly the ACF is decaying, most likely the former.
AIC is helpful, but if you're using it in an automatic fashion, you'll often find a number of models with not much difference in AIC (a difference of less than 2 is often taken as equivalent to "just as good").
In response to the comments:
(1) Is the series stationary or not? It's hard to tell for a short, highly autocorrelated series like this. Unit root tests (KPSS & augmented Dickey-Fuller) might help (but in my experience they rarely tell you anything that isn't obvious from the correlograms & the time series plot itself). A random walk & an AR(1) model with a high AR parameter can both look plausible & pass any diagnostic tests you might perform. Only over the long term are you likely to be able to tell. NB You may have good a priori reasons to pick one or the other.
(2) If it's stationary, AR(1) or more complex model? The ACF hints at other possibilities that are worth testing, but doesn't rule out an AR(1) - remember that real ACFs from short series can look quite different from the theoretical ones. Most people would go for the simplest, at least for the time being, provided that it fits well enough (& see above about AICs). NB A priori considerations can be important here too.
My answer is really an abridgement of javlacelle's but is is too long for a simple comment but not too short to be useless.
While jvlacelle's response is technically correct at one level it "overly simplifies" as it premises certain "things" which are normally never true . It assumes that there is no deterministic structure required such as one or more time trends OR one or more level shifts or one or more seasonal pulses or one or more one-time pulses. Furthermore it assumes that the parameters of the identified model are invariant over time and the error process underlying the tentatively identified model is also invariant over time. Ignoring any of the above is often (always in my opinion !) a recipe for disaster or more precisely a "poorly identified model". A classic case of this is the unnecessary logarithmic transformation proposed for the airline series and for the series that the OP presents in his revised question. There is no need for any logarithmic transformation for his data as there are just a few "unusual" values at periods 198,207,218,219 and 256 which left untreated create the false impression that there is higher error variance with higher levels. Note that "unusual values" are identified taking into account any needed ARIMA structure which often escapes the human eye.Transformations are needed when the error variance is non-constant over time NOT when the variance of the the observed Y is non-constant over time. Primitive procedures still make the tactical error of selecting a transformation prematurely prior to any of the aforementioned remedies. One has to remember that the simple-minded ARIMA model identification strategy was developed in the early 60's BUT a lot of development/improvements have gone on since then.
Edited after data was posted :
A reasonable model was identified using http://www.autobox.com/cms/ which is a piece of software that incorporates some of my aforementioned ideas as I helped develop it. 
The Chow Test for parameter constancy suggested that the data be segmented and that the last 94 observations be used as model parameters had changed over time.
.These last 94 values yielded an equation
with all coefficients being significant.
. The plot of the residuals suggests a reasonable scatter 
with the following ACF suggesting randomness 
. THe Actual and Cleansed graph is illuminating as it shows the subtle BUT significant outliers.
. Finally a plot of actual,fit and forecast summarizes our work ALL WITHOUT TAKING LOGARITHMS 
. It is well known but often forgotten that power transforms are like drugs .... unwarranted usage can harm you. Finally notice that the model has an AR(2) BUT not an AR(1) structure.
Best Answer
Firstly, inferring from the ACF and PACF plots of the data, I would say your series is already stationary. There is no need for first order differencing.
Please refer to the following link for a better understanding on selecting the degree of differencing and order of AR and MA for ARIMA models.
https://people.duke.edu/~rnau/411arim3.htm