Can you reform or explain your question in a different way please ? I am afraid i misunderstood it . Do you want to put the tweeting process under a subject category ? Or do you want each tweet to be categorized according to its subject ?

There is a state-of-the-art in deep learning for short text in SIGIR conference 2016. I think you can check this conference

this depends on type of data you are going to use and tolerance level for  compromising on different requirements. you really need to have a deep look on properties of data then it will be easy to decide any algorithm regarding that.

I think you can start with text classification algorithms, here is a tutorial:

http://www.scientificdatabases.ca/current-projects/english-spanish-text-data-mining/weka-tutorial-on-document-classification/

You can install rattle() on R. It has graphical user interface for support vector machine, regression, normal neural network, random forest for small and medium datasets.

There is not one correct answer since no universal algorithm for any machine learning problem exists. It always depends on your problem and available data.

 Yes, I agree with František. For a specific datasets in a specific parameter settings, you can only said that an algorithm outperforms some other algorithms. Hang Li in Learning to Rank in Information Retrieval book said that the objective for check new machine learning technique in learning to rank is to illustrate the various types of available  machine learning algorithms that can be applicable in learning to rank problem and studying their behavior on that problem domain.

It totally depends on the "subjectivity" of your classification. If you are looking for some classifier for a 2 or 3 class classification, such as "positive", "negative" and/or "neutral" as in the case of Twitter sentiment analysis (which also consists of short length text, 140 chars. to be precise), then there are a wide variety of classifiers available ranging from lexical, machine learning based and hybrid varieties. More info, can be found from a survey paper I wrote quite ago -- https://arxiv.org/pdf/1512.01043.pdf 

I personally used a variant of Naive Bayes classifier for my work on Twitter sentiment analysis (hybrid Naive bayes). You can find more details on the same from my work -- http://harshthakkar.in/wp-content/uploads/2015/06/MTech-2013-Final-Thesis-P11CO010-Sentiment.pdf

There are also various other options available from the deep learning perspective, but these will be rewarding mostly, if you have ample amount of training data. Thus, depending on the subjectivity of your work and the amount of training data you have available, you can choose wisely.

Actually it depends wether your task is to classfiy real stream tweets from the Twitter streaming API in real time manner or to classify tweets from the REST API in offline matter. If you already have a dataset of archived tweets then you can start with classfying work after you do data pre-processign and cleaning and put it in the needed format.

Classification algorithms such as SVM, Naive Bays...etc come in many different formats, some are intended as a immediate way to execute the same algorithms, others might offer a more consistent performance or higher over-all accuracy for the specific problem you have at hand which is classifying tweets into categories. For example if you are currently running your classifier on a Naive Bayes algorithm, then it might be worth considering a Maximum Entropy one or K-mean one.

Thanks

You can also use text mining classificstion methods, you just check in google " R views CRAN TASKS"

https://cran.r-project.org/web/views/

Basically it is a topic classification problem and it has many nuances.  Random Forest and SVM have been giving quite good results. However please note the pre-processing steps, feature selection technique and text representation scheme ( Bag of words, Topic Model, n-gram) will have a bearing apart from the model you are choosing.