Free to Play

Just logging in, not counting events or apologems, you'd get 156 quartz in a year and 52 summon tickets.
Adding in apologems let's take the minimum of 3 per maintenance cycle and you'd get another 50-100 quartz.
Playing through the story would get you roughly 150 quartz. Free quests that come on clearing content in singularities is another 80 quartz roughly at minimum.

That adds up to 430 quartz and 52 summon tickets at minimum just for logging in and doing the quests available all year round or or 195 individual summons (or 14 10 pulls)

None of this even considers events which give additional free stuff, often including event only Welfare SR servants.

The only reason you need to pay is if a limited time event is happening and you just want to up your odds of getting a servant AND (this one is worth paying for) the guaranteed SSR summons because a guaranteed 5* is a guaranteed 5*.

I'm not gonna lie, being f2p in this game is hard for 2 reasons other than being good at making you want to get a servant. 1st the drop rate in this game is pretty low even if you buy quarts if you're not throwing thousands at this game you're not gonna get every 5 star. 2nd the game is pretty selfish with quartz so if you stay f2p don't expect to get more than a couple 5 stars a year if you're lucky. You can still can get some pretty cool well fares for free. For how much quartz well first weekly missions will net around 108 quartz a year and around 200 for daily logins with 20 quartz every 50 days. If you count tickets as well thats another 52 tickets which we will translate as quartz, so without events, maintenance, and the story. Every year you will get around 504 quartz if my math is right, but I would give or take around 20 quartz since i'm not the best at math. This isn't all you will get since the other person neil reminded me that events give quartz and so does maintenance (For a long time I just thought it was daily logins).

I'm F2P I've been plays for almost 4 months and it's not horrible. If you're smart there is a steady trickle of SQ and tickets. You get far more ingame currency than many other mobile games I've played.
In davinci store you get 10 tickets a month plus 1 ticket a week from login bonus.
Every 50 logins you get an extra 20SQ
You get 5 or 6 SQ a week from master missions and daily login
Character interludes, strengthening quest, and bond lvl 6+ on 4,5,6* servants
It all adds up pretty quick. I'm not hitting 10pulls every week. But I've had enough to pull Da Vinci and Ozy from these banners. I do prefer single pulls just in case it is a bad roll I only lose 3SQ vs 30.

F2p as well, and I currently have four 5 star. I don't plan to spend money on the game cause my luck isn't very good, also be sure to buy summon tickets. 218 days I been playing this Gacha game, not that much a big fan of the game anymore. I thought it would be fun the first time, but not anymore.

Simply: 6 Quartz and 1 Ticket (3 Summons) per Week + 50 SQ for first 100 days + 20 SQ for every 50 days after first 100 (150 SQ in year = 50 Summons) = minimum 200 Summons per Year (NOT counting Maintenance Gifts, Event Rewards, Promotional Campaigns). If you believe in RNGod, it should get you at least 2 SSR Servants per year, but I've been playing since February and I got 4 now (F2P). Also a shitload of SR's. Welfare Servants shouldn't be underestimated.

Good Lv 100 10/10/10 Support can carry you through roughly half of every event. Events are quite common (save Apples for them) and can grant you tons of mats, apples, CE's and Summons (depending on event, I'd say about 1 Summon every day through whole event, granted if it's event with separate map). F2P players are not really at disadvantage (this game contains NO PvP system so we are all commies when it comes to OUR servants...). I think you should keep playing, don't waste energy (after you finish free quests go for Ember Gathering, only 40 AP ones), in event shops aim for mats, gems and monuments).