I'm not well enough informed to say anything certain about what sort of authorities you could go to for help and how the potential process of investigation and prosecution would work, but that's a pretty interesting question. It would certainly dependent where in Rome you lived (in cities or countryside, city of Rome or a provincial town etc.), as the administrative structures could differ a lot. Also, your own social status would play a big role on what sort of help you could get; a rich senator certainly would have more options than a poor fisherman. I'm not really in a position (i.e. on a holiday with no library) to research this and say anything more, hopefully someone else will answer this.

I thought, however, that I could chip in with the practice of the so-called curse tablets, the use of which you might have seen depicted in the HBO/BBC Rome series. This is a method of achieving 'justice' characteristic, although not exclusive, to the lower classes, who probably didn't have the money and influence to get much through the official routes. The curse tablets, also known as defixiones in the academic lingua, were basically inscribed lead tablets intended to bring supernatural powers against persons or animals. The tablets were usually deposited to somewhere with a good 'connection' with the gods or the victim, such as under the victim's house or to a spring sacred to the called-upon god or goddess.

The inscriptions are often quite formulaic, sometimes include 'magic words', or 'pig-latin' presumably intended to be language only gods could understand, and also can be quite inventive in what horrendously painful punishments were plotted to fall upon the victim. Although, it might be sensible to take these metaphorically rather than literally. Some curse tablets were also used to 'fix' races, to bring accidents on racehorses, but also to try to force people to fall in love with the writer. The desire for revenge and justice for theft is one motive for inscribing a curse tablet, and this type is especially common in Britain. In these, the owner of lost property would plead do a divinity or divinities to bring down punishment or reveal identify the thief; and quite often a list of names of the suspected potential thieves is added. Here's a couple of amusing examples (these are all from Imperial Roman Britain as that's the sample I'm familiar with, but you could find very similar ones elsewhere in the empire during the Republican times, too):

. . . in 3 thousand denarii, of which (I give) you half part on condition that you exact it from Vassicillus the son of [. . .]cominus and from his wife, since the coin(?) which they have stolen(?) from my house (the syntax breaks down) and you are not to permit them health nor to drink nor to eat nor to sleep [nor] to have healthy [children] unless they bring this [my] property to your temple. With repeated [prayers] I ask you that this [coin(?)] may come to be recovered [from the very] names of my [enemies].

Pagans Hill A (no. 7) Hassall and Tomlin 1984, 339.

Biccus gives Mercury whatever he has lost (that the thief), whether man or male (sic), may not urinate nor defecate nor speak nor sleep nor stay awake nor [have] well-being or health, unless he bring (it) in the temple of Mercury; nor gain consciousness (sic) of (it) unless with my intervention.

Tomlin (no. 2), Britannia 19, 1988 485-6

Lord Neptune, I give you the man who has stolen the solidus and six argentioli of Muconius. So I give the names who took them away, whether male or female, whether boy or girl. So I give you, Niskus, and to Neptune the life, health, blood of him who has been privy to that taking-away. The mind which stole this and which has been privy to it, may you take it away. The thief who stole this, may you consume his blood and take it away, Lord Neptune.

Hassall and Tomlin (no. 1) 1997, 455-58

May he who carried off Vilbia from me become as liquid as water. (May) she who obscenely devoured her (become) dumb, whether Velvinna, Exsupereus, Verianus, Severinus, A(u)gustalis, Comitianus, Catus, Minianus, Germanilla (or) Jovina.

Bath, RIB 154 (Vilbia might have either been a female or perhaps an object not otherwise known in latin, such as a bath towel)

As ridiculous and superstitious as this habit might sound, the fact that curse tablets were in use for a good 1,000 years if not a lot longer (Romans got the custom from the Greeks) clearly means that they did something. Even if the thief didn't turn into liquid in front of your eyes, it at least had psychological benefits: cursing a person removed tension and allowed for a transfer of emotions. At least something had been done, and even if the thief might not suffer immediately, you could foster the belief that the curse would get to them eventually. As R.S.O. Tomlin has noted, the modern categories of 'law', 'magic', and 'religion' aren't very useful when trying to understand curse tablets; the tablets use formulas and wording that can be called 'quasi-legal'. Turning to gods for justice was just as logical to the ancients as calling for police is for us; and perhaps supernatural powers were seen as more reliable than corrupted and snobbish Roman officials. So, if your house got robbed in Republican Rome, getting a curse tablet which stopped the thief from urinating etc. might be a good shout.

For further reading, especially recommend R.S.O. Tomlin's (1988) report on the Curse Tablets from the Temple of Minerva Sulis in Bath, and J. Gager's book (1992) Curse Tablets and Binding Spells from the Ancient World. Here's also a catalogue of a small sample of curse tablets found from Roman Britain.

How about in the first ceasars period compared to Caligulas period. When did social services come about? Nero had city firefighters, right?

That's a very broad time period, you mean early Rome? Julius Caesar's period? Caligula's Rome? Rome when the Western Roman Empire fell?