[Hum, this turned into a longer wall of text than I expected x) You can add me ingame if you want to chat ;) ]
Lol, I'm kinda flattered. I checked your profile and I see that you're in Plat right now when I'm just in Gold, and I'm preaching a play style that is very different from the "common wisdom" about Sona, so I would've expected a dismissal ^^'
But yeah, I honestly think that the apparent synergy between Sona's kit and Lich-bane is only a trap and that there are far better items for her.
If you want more details: I generally start spellthief + mana regen charm, rarely with pink ward + 1 potion. The mana regen charm + the mana regen mastery allows to basically spam heal in lane if you need, against a hard lane, so no need to take potions. The pink is against stuff like shaco/eve.
Your basic pattern in lane is: close in to poke with your blue and shield the retaliation with your green -- but only if both you and your ADC are not full life, it's important to optimize your mana efficiency. If you/your ADC make a mistake and take a bad trade / are chunked for a large portion of your life, slow down on the poke and keep mana for heal. If you spend at least the same amount of mana for your green spell as for your blue, take 2 points in each, otherwise max blue. Usually, try to put a point in pink level 4 or 5, but I'm often greedy and put another point in green/blue and only take pink level 7 or 8.
Buy gold item upgrade and sightstone ASAP, and basic boots, but you don't really need T2 boots, and you don't really need to combine the gold item and the sightstone until you need to free the slot. If you have 125g left after a back (including buying a pink ward ;) ), buy a faery charm (you need 4 of them in total, so you'll always find something to do with it). Once you have gold item/SS, you can move on to mid-tiers components. If you're ahead, Codex is great; if you need mana regen, Idol is great; if you are behind in your lane, Chalice is OK; and sometimes you need to rush an Aegis if the midlaner is snow-balling and roaming a lot.
After that, complete the 3 items in whatever order seems appropriate (Censer/Athene's/Aegis upgrade). If I can get away with it, I really love Banner of Command (I also build ZZ'rot on mots top laners when I go top), but sadly the team really needs a Solari Locket more often than not.
Once teams start to group, you can poke with your blue if it's safe, but it's no longer your top priority. Instead, focus on: 1/ vision, vision, vision; 2/ spamming your auras in team-fights, don't try to get near the carries to hit them with your blue, it's not worth the risk, your main "damage output" will be your blue aura and your green aura applying the Censer passive to your whole team; 3/ wave manipulation, especially if you bought Banner. If you didn't buy Banner, you cannot execute wave manipulation yourself (Sona has absolutely no wave-clear, it's her main weakness in Mid), but it's still your job as the support to watch the minimap and to look for opportunities and to communicate them to your carries.
(Digression about the "damage output": Ardent Censer is even better on Sona than on Soraka. You can basically give a permanent +15% AS to your whole team, +30 on-hit damage. It's much better than it looks, almost as powerful as a reversed Frozen Heart Aura, and especially with stuff like Yi/Vayne/Kog'maw/Warwick/Shyvana/Jax/Irelia/etc. Vayne and Kog also love laning with a Sona who can basically sustain them to infinity while they farm, and distract the enemies with her poke.)
At this point, you need to combine sightstone and frostfang to free up your last slot, and you can move onto pretty much whatever you want, but keep in mind that you need at least 250 AP total for Lichbane's procs to be better than Luden's, and Luden also gives more raw AP to improve your heal/shield/Speed buff, and more %MS. Sometimes I go for Randuin or Zhonya if I want Armor, sometimes Frozen Heart if the enemy team has lots of AS based champs (but you don't need the mana and you're wasting the CDR, so it's only if the team really, REALLY needs the aura). Anyway, we're talking 6th item for a Support, so it's pretty rare.
**Elixir: **blue is better in 95% of cases, but once in a while, you can take the orange for the 300 HP + tenacity.
**Boots: **I usually upgrade them last, even after the elixir. I'm already at 45% CDR so I don't want CDR boots, and in late-game I'm not here for damage so I don't really care about Sorc, especially given that your blue aura benefit from your allies penetration, not your own. Nothing is clearly good, so you can take whatever you want for this specific game. Defense boots if you die too fast, Mobility if you want to roam a lot to ward the map (but you have your pink spell anyway), and sometimes Sorc if you don't want anything else. Two exceptions: if we crushed our lane and I want to roam a lot in early, I take mobility boots, and if we are snow-balling and I want to double down on that, I take Sorc, because that's the best/cheapest way to boost your damage without investing in heavy AP items.