How do I find my weight loss per week, month, etc.? I want to statistically find out how much weight I am losing at different time intervals (every week, month, 2 months, etc.). What statistics should I be performing? Paired t-test? Regression? 
I'm weighing myself every day in the morning and it varies day-by-day. For example, I can weigh higher in one day and lower the next or vice versa. I feel that a simple subtraction after a week (7 days) or after a month (~30 days) doesn't account for variability nor give me the confidence interval. I wouldn't know unless I was able to determine with statistics how my weight loss looks at those time intervals. 
 A: By weighing yourself at the same time each day, you are improving the usefulness of the data by controlling for within-day variations.  But such variations will continue to occur, because consumer scales are horribly imprecise and the body retains more or less fluid seemingly at random each day.
Since this is a simple question for a simple study, it deserves a simple practical answer: consider smoothing the data.  Use the smoothed values instead of the actual values to estimate changes during any interval simply by subtracting the end weight from the beginning weight.

Smoothing is a bit of an art, but you can do a good job by plotting the data and using good judgment.  Let me illustrate with data.  Those plotted in this post were obtained by regular weighings with the same consumer scale at the same time of day, adjusted for the weight of any clothing.  I have shifted the times (and the weights) so that we may focus on the general issue rather than the specific numbers shown; but otherwise, these are real data.  (The weights are in pounds.)
Like all real data, they are imperfect.  In particular, although nominally the weighings occurred every day, sometimes the weight is missing for various reasons unconnected with the weight.  We may take these values to be missing completely at random--we won't worry about them.  But this irregular timing is one reason not to use sophisticated time-series methods for the analysis.

These plots all show the same series of data, shown as gray points.  The plots differ in how aggressively the smoothing was done.  The "span" in the key at right is, roughly, the fraction of all points contributing to the smoothed value at any one point.  Only neighboring point values were used, weighted so that nearer points contribute more than further points.  Larger spans will smooth more.  Roughly, for this 1.25 year period, "3/5" shows semiannual trends, "1/4" shows quarterly trends, "1/10" shows monthly trends, and "1/24" shows biweekly trends.
Although a span of 3/5 doesn't track the data consistently, it does show the overall picture: weight fell steadily for a half-year and then leveled off.  You might use it to make long-term comparisons (for intervals of a year or more).
The span of 1/4 looks like it follows all the obvious big trends.  On top of the annual pattern shown at its left, it appears to document a seasonal cycle of a slight (2.5 pound) weight gain in the spring and autumn compared to winter and summer.
Details difficult to detect by eye show up with a span of 1/10, but they don't seem to change things much compared to the span of 1/4: their amplitude is about a pound.  For tracking weight changes over periods of one month or more, this or the preceding smooth might be the best choices.
The smooth with a span of 1/24 probably shows more detail than these data can support. 
How can we know the last plot may be showing too much detail? By looking at the residuals, which are the errors made by replacing the raw data by their smooths.  Formally: the residual is the difference between the raw weight and its smoothed value for that date.

The residuals for a span of 3/5 show patterns: they clearly trend above and below zero (their ideal value).  Many of those residuals are shockingly large, several pounds or more.  That's terrible for a scale that reads in units of 0.1 pounds!  However, it incorporates not just measurement error but also random, uncorrelated, variable changes in the body itself, such as how much fluid it is retaining. Such patterns largely disappear well before the span reaches 1/24, with the large majority of residuals less than a pound.  This fluctuation is probably "noise"--random, uncorrelated measurement error and irreducible variation.
(Periods of smaller residuals are evident in bottom plots around May 2020 and February 2021, suggesting there may be periods in which such variable body conditions changed less from day to day than otherwise.)
If we were to smooth even less--and therefore follow the data even more closely--the residuals would continue to shrink towards zero, but the smooth would merely be tracking the random noise rather than following any real underlying signal you want to capture.
How can you figure out at what point to stop smoothing?  Roughly, where the amplitude of the residuals begins to shrink appreciably.

To make this plot, I made 11 smooths with spans ranging from 1/48 through 2/3.  For each I computed the spread of the residuals as their median size: half the residuals are greater in size than the spread and half are less than it in size.
This plot shows a clear "elbow" near a span of 0.1 (1/10).  This is where smooths with smaller spans follow the individual data values so closely that the residuals rapidly diminish.  We can interpret the gradual slope for spans between 1/10 and 2/3 as reflecting the variations in true weights being picked up as the span is varied in this range.  The steep slope for spans between 1/48 and 1/10 reflects a different smoothing regime in which the smooth begins to be controlled by the measurement error.

Although you can smooth data manually, software makes it easy.  All these smooths were instantly computed by the loess function in the R statistical package.  (Many other statistical packages have a version of loess.)
(Note that the loess "span" is a fraction of all the data.  Its meaning therefore will change as your dataset grows.  You cannot assume the right span for your data will be the same fraction as the best one found here (1/10), nor that it will remain constant while you continue monitoring.  You have to analyze your own data.)
You don't have to be this fussy in choosing your smooth.  Experiment with the first month or two of data.  Don't track the data too closely.  Smooth just enough to capture the visual trends you can see.  Then continue to smooth at that level while you continue to monitor the weight.  
A: Maybe exponential smoothing is helpful. It levels out day-to-day variability, and weights recent observations more than older ones. 
A: I have the app that measures my weight every day at the same time. I don't do any stat analysis. It's pointless. Just draw the plots as they appear over time, you'll see what's going on. The only thing to do would be the trend extraction, which can be done by a simple or exponential moving average. Whatever you do should be non-parametric, because there's no model of your weight.
You can try the intervention study if you're trying a different diet, but it's very difficult to do properly. The main reason is seasonality and other stuff going on in your life (which is called controls in econometrics), such as aging.
The seasonality is a the beast. I've been drawing my weight for years, and can attest that holiday weight gain (around thanksgiving) is a real phenomenon :) 
Aging is huge too, plus other peculiarities of your life style, vacations, big events, stresses etc. You need to control for all of this in order to do statistical intervention analysis.
However, a simple time plot of the weight with filters such as EWMA gives you all that is necessary. You'll see if your weight is decreasing or not once there's enough data collected.
