The major difference between reconstructive religions and modern ones--especially Pagan ones--is the way worship is conducted. Individual worship of Gods as well as patronage is perfectly acceptable in modern religions, but in Recon religions and the ancient Traditions they were based upon, worship tends to be of the pantheon, not so much the one God or Goddess. As such, it is hard for me to answer questions like 'how should I worship Athena', or 'what do I do to honour Poseidon'; you do pretty much the same for both, and all other Olympic Gods as well--at least at it's core. The devil is in the details, but that is beyond the scope of this post.
I have mentioned before that there are five steps to proper, Hellenistic, ritual: procession, purification, prayers and hymns, sacrifice/offerings, prayers of supplication and thanks, usually followed by a feast and/or theater and sporting events. We can apply this to modern worship quite easily: procession (no matter how short), purification with lustral water (named khernips), a hymn, song or modern poem which praises and draws the Theos in question, a sacrifice of some kind--be it incense, (mixed) wine, meat or anything else--along with barley seeds tossed on the altar or into the altar fire, prayers or words of thanks, and--in communal rituals--plays, games, or (sports)-competitions. Within communal celebrations, the sacrifice can be some of the (raw) ingredients used to prepare the communal meal that will follow.
For those of you looking to honour the Gods with sacrifices that are not the above staples, make sure you are aware of the mythology surrounding the God or Goddess in question. In general, any sacrifice is acceptable--although probably not Traditional--but some plants/trees/fruits can be either extra special to the God in question (Laurel for Apollon, for example), or extra painful (pomegranate seeds for Demeter, for example).
We have great variety in ancient hymns that you can draw from to worship just about any deity; it's important, though, that in the context of the worship of one God or Goddess, you at least include those closest to Him or Her. In the case or Apollon, for example, don't forget His sister Artemis, His Mother Leto, His father Zeus, and Hera in placation. For Athena, don't forget Zeus, Ares, and Hera, and for--for example--Poseidon, don't forget Zeus, Amphitrite, and Rhea. Kronos if you are so inclined. Just never forget Zeus, ever. Also, in general, Hellenists offer to Hestia first and last.
Part two: shrines. An important note first: there is a huge difference between an altar and a shrine. An altar is one of those basic necessities within Hellenismos, and it differs from a shrine. Where an altar is a 'work space', dedicated not so much to a specific deity, but used to do the bulk of the (daily) rituals, a shrine is a devotional area where an altar might be located. In ancient Hellas, the shrine was usually a temple, the altar an actual altar, standing outside of it. Household worship took place at a multitude of shrines. Labelling something a shrine, does not mean you can't sacrifice at these spots in your home. In general, you decorate a shrine but leave the altar rather bare.
As for decorating your shrine: delve into mythology and go as wild as you want. In general, a light source and an offering bowl are staples, the rest is up to you. For Poseidon, the great Olympian God of the sea, rivers, flood and drought, earthquakes, and horses, you might look for the trident that is His symbol, appeased with fumigations of myrrh incense and the outpour of (sea) water. You could add images or statues of horses, Himself, and perhaps His wife Amphitrite to your shrine, and add seashells and anything else you can collect off of a beach for decoration. For Athena, I think owls would be a staple, Her weapon is the spear, anything with olives (including an olive tree) would be fantastic, etc.
I'm going to let you in on a little secret: Hellenismos is not glamorous; in general, you do the same thing over and over again with minor variations. That is what I love about it. It's simple, clear, and repetitive. Practice it like that and it'll become engrained into your person, and the Gods will become part of your daily life. That is the beauty of reconstruction. Remember this when you do your rituals and make up your shrines: practicality is a great good!