How to estimate the probability that the next observation in a time series will be bigger than 0? I have time series data and I want to know how to find out the probability that the next one will be bigger than zero. This time series doesn't have a seasonal aspect to it but it does have a slope and a level and autocorrelation.

This is a very typical task in financial risk management: forecast portfolio value distribution for next period, then analyze its characteristics such as quantiles.
Here's how it's done. You fit the time series model to the data set of portfolio values $p_t$ or returns $r_t=\Delta\ln p_t$, then produce a probability distribution $\hat F(p_{t+h}|I_t)$ of the portfolio values $p_{t+h}$ for next period given what you know today $I_t$. Once you know the distribution, you can trivially obtain probabilities such as of below zero $Pr[p_{t+h}<0]=\hat F(0|I_t)$
The distributions can be obtained by bootstrapping but sometimes are available with analytics expressions, e.g. if you're using ARIMA type of models. The general form of ARIMA is:$$\Phi(B) \Delta^dx_t=\Theta(B)\varepsilon_t$$
You usually make some kind of distributional assumption about $\varepsilon_t$ estimating the model, e.g. $\varepsilon_t\sim\mathcal N(0,\sigma^2)$. This means that you can simulate (sample from) normal distribution to get these disturbances then plug them back into the model to produce the sample from forecast $\hat x_{t+h}$.
